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Interacting particle systems, spectral gap,
Poincare inequality, Log-Sobolev inequality, Conservative spin systems,
Continuous spin systems
---------------0201210008946
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Content-Transfer-Encoding: 7bit
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\newcommand{\mykeywords}{Interacting particle systems, spectral gap, 
Poincaré inequality, Log-Sobolev inequality, Conservative spin systems, 
Continuous spin systems}
        
\newcommand{\mysubjclass}{
60K35 Interacting random processes, 
82B44 Disordered systems, 
82B20 Lattice systems,
46-99 Functional analysis,
60J60 Diffusion processes,
26D10 Inequalities involving derivatives, differential and integral operators}

\title{Glauber versus Kawasaki for spectral gap\\ and logarithmic Sobolev
  inequalities\\ of some unbounded conservative spin systems}

\date{University of Toulouse\\ \medskip \small \today}

\author{D.~Chafaï}

%
\newcommand{\MSX}{\EM{M-\sum_{i=1}^n x_i}}
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%

\begin{document}

\maketitle

\begin{abstract}
  Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau
  \cite{landim-panizo-yau} on spectral gap and logarithmic Sobolev
  inequalities for unbounded conservative spin systems, we study uniform
  bounds in these inequalities for Glauber dynamics of Hamiltonian of the form
  $$
  \sum_{i=1}^n V(x_i)+V\biggl(\MSX\biggr), \quad (x_1,\ldots,x_n)\in\dR^n
  $$
  Specifically, we examine the case $V$ is strictly convex (or
  small perturbation of strictly convex) and, following
  \cite{landim-panizo-yau}, the case $V$ is a bounded perturbation of
  a quadratic potential. By a simple path counting argument for the
  standard random walk, uniform bounds for the Glauber dynamics
  yields, in a transparent way, the classical $L^{-2}$ decay for the
  Kawasaki dynamics on $d$-dimensional cubes of length $L$. The
  arguments of proofs however closely follow and make heavy use of
  the conservative approach and estimates of \cite{landim-panizo-yau},
  relying in particular on the Lu-Yau martingale decomposition and
  clever partitionings of the conditional measure.
\end{abstract}

%
\section*{Introduction}
%

Let $Q$ be a probability measure on $\dR^n$. In the sequel, we denote
by $\moy{Q}{f}$ the expectation of $f$ with respect to $Q$,
$\var{Q}{f}:=\moy{Q}{f^2}-\moy{Q}{f}^2$ the variance of $f$ for $Q$,
and $\ent{Q}{f}$ the entropy of a non negative measurable function $f$
with respect to $Q$, defined by
$$
\ent{Q}{f}:=\int\! f\log f dQ - \int\! f dQ \;\log \int\! f dQ.
$$
We say that $Q$ satisfies a Poincaré inequality if there exists a positive
constant $\cP$ such that for any smooth function $f:\dR^n\to\dR$,
\begin{equation}\label{in:def-ts}
\var{Q}{f} \leq \cP\, \moy{Q}{\ABS{\GR f}^2},
\end{equation}
where $\ABS{\GR f}^2:=\sum_{i=1}^n \ABS{\pd_i f}^2$. Similarly, we say that
$Q$ satisfies a logarithmic Sobolev inequality if there exists a positive
constant $\cL$ such that for any smooth function $f$,
\begin{equation}\label{in:def-ls}
\ent{Q}{f^2} \leq \cL\, \moy{Q}{\ABS{\GR f}^2}.
\end{equation}
This inequality strengthen the Poincaré inequality \eqref{in:def-ts} since for
$\veps$ small enough,
$$
\ent{Q}{(1+\veps f)^2}=2\veps^2\var{Q}{f}+\cO(\veps^3),
$$
which gives $2\cP \leq \cL$. Let $\cH\in\cC^2(\dR^n,\dR)$ such that
$$
Z_{\cH}:=\int_{\dR^n}\!\!e^{-\cH(x)}\,dx < +\infty.
$$
The probability measure $Q$ defined by
$dQ(x)=(Z_{\cH})^{-1}\exp\PAR{-\cH(x)}\,dx$ is the symmetric invariant measure
of the diffusion process $(X_t)_{t\geq 0}$ on $\dR^n$ driven by the S.D.E.
$$
dX_t = \sqrt{2}\,dB_t - \GR \cH(X_t)\,dt,
$$
where $(B_t)_{t\geq 0}$ is a standard brownian motion on $\dR^n$.
In this context, we say that the probability measure $Q$ is associated
with the ``Hamiltonian'' $\cH$. It is well known that $Q$ satisfies
the Poincaré inequality \eqref{in:def-ts} with a constant $\cP$ if and
only if the infinitesimal generator $\GI:=\LA-\GR\cH\cdot\GR$
possesses a spectral gap greater than $\cP^{-1}$. In the other hand, a
famous Theorem of Gross states that $Q$ satisfies the logarithmic
Sobolev inequality \eqref{in:def-ls} if and only if the diffusion
semi-group generated by $\GI$ is hyper-contractive. A celebrated
result of Bakry and Émery ensures that when there exists a constant
$\rho>0$ such that for any $x\in\dR^n$,
$$
\Hess{\cH}(x)\geq \rho\,\bI_{p}
$$
as quadratic forms on $\dR^n$, i.e. $\cH$ is uniformly strictly convex or
$Q$ is log-concave, then $Q$ satisfies to \eqref{in:def-ts} and
\eqref{in:def-ls} with constants $\cP=\rho^{-1}$ and $\cL=2\,\rho^{-1}$
respectively. Moreover, $Q$ satisfies to \eqref{in:def-ts} with a constant
$\cP$ if and only if
$$
\cP\moy{Q}{(\GI f)^2} \geq \moy{Q}{\ABS{\GR f}^2}
$$
for any smooth function $f$. The reader may find an introduction to
logarithmic Sobolev inequalities and related fields in \cite{our-ls-2000}.
 
We are interested in the present work to particular ``Hamiltonians''
$H\in\cC^2(\dR^{n+1},\dR)$. Let $M\in\dR$ and define $H_M\in\cC^2(\dR^n,\dR)$
by
\begin{equation}\label{eq:defhm}
  H_M(x_1,\ldots,x_n):=H\biggl(x_1,\ldots,x_n,\MSX\biggr).
\end{equation}
Assume that $Z_{H_M}<\infty$. Our aim is to establish Poincaré and
logarithmic Sobolev inequalities for probability measures on $\dR^n$ of the
form
\begin{equation}\label{eq:defnum}
d\si_M(x_1,\ldots,x_n):=(Z_{H_M})^{-1}\,\exp\PAR{-H_M(x)}\,dx_1\cdots dx_n,
\end{equation}
with constants $\cP$ and $\cL$ which does not depend on $n$ and $M$.  This
investivation is motivated by the study of certain conditional probability
measures. Namely, if the probability measure $\mu$ on $\dR^{n+1}$ given by
\begin{equation}\label{eq:defmu}
d\mu(x):=(Z_H)^{-1}\,\exp\PAR{-H(x_1,\ldots,x_{n+1})}\,dx_1\cdots dx_{n+1}
\end{equation}
is well-defined, i.e. $Z_H<+\infty$, then for any $M\in \dR$, one can define
the conditional probability measure $\mu_M$ by
\begin{equation}\label{eq:defmum}
\mu_M:=\mu\biggl(\;\cdot\;\Bigr\vert\;\sum_{i=1}^{n+1} x_i=M\biggr),
\end{equation}
and we get, for any $f\in\cC_b(\dR^{n+1},\dR)$,
\begin{equation}\label{eq:mum2num} 
  \moy{\mu_M}{f}=\int_{\dR^n}\! f\biggl(x_1,\ldots,x_n,\MSX\biggr)\,
  d\si_M(x_1,\ldots,x_n). 
\end{equation}
Thus, $\si_M$ can be viewed as the translation of the conditional
probability measure $\mu_M$ under the affine hyperplane of $\dR^{n+1}$
of equation $x_1+\cdots+x_{n+1}=M$. Alternatively, and following
Caputo in \cite{caputo}, the conditional probability measure $\mu_M$
can be defined from the probability measure $\mu$ given in
\eqref{eq:defmu} by adding an infinite potential outside of the affine
constraint $x_1+\cdots+x_{n+1}=M$. Namely for any bounded continuous
function $f:\dR^{n+1}\to\dR$
$$
\moy{\mu_M}{f}=\lim_{\be\to+\infty} \moy{\mu_{M,\be}}{f}.
$$
where $\mu_{M,\be}$ denotes the probability measure on $\dR^{n+1}$
defined by
$$
d\mu_{M,\be}(x) 
:= \cZ_{\mu_{M,\be}}^{-1}\,e^{-\be\,\PAR{M-x_1-\cdots-x_{n+1}}^2}\,d\mu(x).
$$
A simple change of variable in $\moy{\mu_{M,\be}}{f}$ gives that
$$
\lim_{\be\to+\infty} \moy{\mu_{M,\be}}{f}=
\int_{\dR^n}\! f\biggl(x_1,\ldots,x_n,\MSX\biggr)\,d\si_M(x_1,\ldots,x_n),
$$
This weak limit definition of $\mu_M$ was used by Caputo in
\cite{caputo} in order to study the case of a convex Hamiltonian $H$.
We do not use it in our approach.  Notice that if
$f\in\cC_b(\dR^n,\dR)$, we get from \eqref{eq:mum2num} that
\begin{equation}\label{eq:mumeqnum} 
  \moy{\si_M}{f} 
  =\int_{\dR^{n+1}}\!f(x_1,\ldots,x_n)\, d\mu_M(x_1,\ldots,x_{n+1}).
\end{equation}
Observe that \eqref{eq:mum2num} gives $\moy{\mu_M}{x_1+\cdots+x_{n+1}}=M$.
Thus, when $H$ is a symmetric function, $\si_M$ and $\mu_M$ are exchangeable
measures, i.e. invariant by any permutation of the coordinates. This holds for
example when $H(x)=V(x_1)+\cdots+V(x_{n+1})$.  Moreover, $\MSX$ and $x_j$ have
then the same law under $\si_M$ for any $j$ in $\{1,\ldots,n\}$ and we get
\begin{equation}\label{eq:mum-num-moy}
  \moy{\mu_M}{x_1}=\cdots=\moy{\mu_M}{x_{n+1}}
  =\moy{\si_M}{x_1}=\cdots=\moy{\si_M}{x_n}
  =\frac{M}{n+1}.
\end{equation}
Thus, the mean of $\mu_M$ and $\si_M$ does not depend on $H$ in this case.

Let us see now how to translate \eqref{in:def-ts} and \eqref{in:def-ls} for
$\si_M$ in terms of $\mu_M$. One can observe that for any $i\in\{1,\ldots,n\}$
\begin{multline*}
  \pd_i\BPAR{f\BPAR{x_1,\ldots,x_n,\MSX}} \\
  =(\pd_i f)\BPAR{x_1,\ldots,x_n,\MSX}-(\pd_{n+1} f)\BPAR{x_1,\ldots,x_n,\MSX}.
\end{multline*}
By replacing the coordinate $x_{n+1}$ by any of the $x_1,\ldots,x_n$ in
\eqref{eq:mum2num}, we obtain the following proposition
\begin{prop}\label{pr:num-mum}
  Let $H:\dR^{n+1}\to\dR$ and assume that for any permutation $\pi$ of the
  coordinates, the probability measure $\si_M^\pi$ on $\dR^n$ defined by
  \eqref{eq:defnum} and associated to $H\circ \pi$ satisfies to Poincaré
  (resp.  logarithmic Sobolev) inequality with a constant $\cP$ (resp. $\cL$)
  which does not depend on $n$, $M$ and $\pi$.  Then, if $\mu_M$ is the
  associated conditional probability measure defined by \eqref{eq:defmum}, we
  get for any smooth $f:\dR^{n+1}\to\dR$
  \begin{equation}\label{in:ts-ij}
   \var{\mu_M}{f} 
   \leq 
   \frac{\cP}{n+1}
   \;\moy{\mu_M}{\sum_{1\leq i,j\leq n+1}\ABS{\pd_i f-\pd_j f}^2},  
  \end{equation}
  and respectively
  \begin{equation}\label{in:ls-ij}
  \ent{\mu_M}{f^2} 
  \leq 
  \frac{\cL}{n+1}
  \;\moy{\mu_M}{\sum_{1\leq i,j\leq n+1}\ABS{\pd_i f-\pd_j f}^2}.
\end{equation}
\end{prop}
These inequalities leads to constants in $L^2$ for the ``Kawasaki
dynamics'' associated to $\mu_M$. Namely, consider a finite box
$\La:=\{1,\ldots,L\}^d\subset\dZ^d$ on the lattice $\dZ^d$ and $n$
such that $\dR^\La\simeq\dR^{n+1}$ (i.e. $n+1=\ABS{\La}=L^d$). There
exists a constant $C>0$ depending only on $d$ such that for any
$a\in\dR^\La$
\begin{equation}\label{in:compdirf} 
 \frac{1}{\ABS{\La}}\sum_{i,j\in\La}(a_i-a_j)^2
 \leq 
 C\;L^2\;\sum_{\SSK{i,j\in\La\\\ABS{i-j}=1}}(a_i-a_j)^2,
\end{equation}
Therefore, it is straightforward to deduce from \eqref{in:ts-ij} and
\eqref{in:ls-ij} that for a constant $C>0$ which does not depend on
$n$ and $M$, one have
\begin{equation}\label{in:ts-l2}
\var{\mu_M}{f} 
\leq C\,L^2\; \sum_{\SSK{k,l\in\La\\\ABS{k-l}=1}}
\moy{\mu_M}{\ABS{\pd_i f-\pd_j f}^2},
\end{equation}
and
\begin{equation}\label{in:ls-l2}
\ent{\mu_M}{f^2} \leq
C\,L^2\;
\sum_{\SSK{k,l\in\La\\\ABS{k-l}=1}}
\moy{\mu_M}{\ABS{\pd_i f-\pd_j f}^2}.
\end{equation}
Inequality \eqref{in:compdirf} follows from a classical path counting argument
(see for example section 4.2 of \cite{saloffcoste}). However, let us gives
briefly a proof. For any $i,j$ in $\La$, consider the path $\Ga_{ij}$ inside
$\La$ joining $i$ and $j$ obtained by adjusting the $d$ coordinates one after
the other. We have $\ABS{\Ga_{ij}}\leq dL$ and for each $k,l$ in $\La$ with
$\ABS{k-l}=1$, the number of such paths containing the edge $(k,l)$ is bounded
above by $c_d\,L^{d+1}$ where $c_d>0$ is a constant depending only on $d$. Now
by Cauchy-Schwarz's inequality
$$
(a_i-a_j)^2
=
\SBRA{\sum_{(k,l) \in \Ga_{i,j},\,\ABS{k-l}=1}\!\!(a_k-a_l)}^2
\leq
dL \sum_{(k,l) \in \Ga_{i,j},\,\ABS{k-l}=1} (a_k-a_l)^2,
$$
and therefore
$$
\sum_{i,j\in\La} (a_i-a_j)^2
\leq
dL \sum_{\SSK{\ABS{k-l}=1\\k,l\in \La}}
(a_k-a_l)^2\sum_{\SSK{i,j\in\La\\\Ga_{i,j}\ni(k,l)}}\!\!\!1
\leq 
dc_d L^{d+2} \sum_{\SSK{\ABS{k-l}=1\\k,l\in \La}} (a_k-a_l)^2,
$$
which gives the desired result \eqref{in:compdirf}.

A simple example is given by uniformly strictly convex $H$ with a constant
$\rho$ in $\dR^{n+1}$. Namely, if there exists a constant $\rho>0$ such that
for any $x\in\dR^{n+1}$, $\Hess{H}(x)\geq \rho \bI_{n+1}$ as quadratic forms
on $\dR^{n+1}$, then, an easy calculus gives for any $x\in\dR^n$ and
$h\in\dR^n$
$$
\DP{(\Hess{H_M})(x) h,h}_{\dR^n}
\geq \rho
\sum_{i=1}^n h_i^2 + \rho \PAR{-\sum_{i=1}^n h_i}^2 \geq \rho \sum_{i=1}^n h_i^2.
$$
Thus, $H_M$ is uniformly strictly convex with the same constant
$\rho$, and therefore, by the Bakry-Émery criterion, $\si_M$ satisfies
to Poincaré and logarithmic Sobolev inequalities with a constant
$\rho^{-1}$ and $2\,\rho^{-1}$ respectively, which does not depend on
$n$ and $M$.  The hypotheses of Proposition \ref{pr:num-mum} are
full-filled since by the same calculus, $(H\circ\pi)_M$ is also
uniformly strictly convex with a constant $\rho$. A more simple
example is given by 
$$
H(x)=V(x_1)+\cdots+V(x_{n+1})
$$
where $V$ is in $\cC^2(\dR,\dR)$ with $V''>\rho>0$.  Let us
consider now another convex Hamiltonian example on $\dR^{n+1}$ defined
by
$$
H(x):=\frac{1}{2(n+1)}\,\sum_{i,j=1}^{n+1} V_{\{i,j\}}(x_i-x_j),
$$
where $V_{\{i,j\}}$ are in $\cC^2(\dR,\dR)$ and even.  This is a so called
mean-field Hamiltonian when all the $V_{\{i,j\}}$ are equal.  We have for any
$i,j$ in $\{1,\ldots,n+1\}$
$$
(n+1)\,\pd^2_{ij} H(x)=
\begin{cases}
  \sum_{\SSK{k=1\\k\neq i}}^{n+1} V''_{\{i,k\}}(x_i-x_k) & \text{ if } i=j\\
  -V''_{\{i,j\}}(x_i-x_j) & \text{ if } i\neq j
\end{cases}
$$
Therefore, if $V''_{\{i,j\}}(u)\geq 0$ for any $u\in\dR$ and any
$i,j\in\{1,\ldots,n+1\}$, i.e. $V_{\{i,j\}}$ is convex, the
Gershgorin-Hadamard theorem implies that for any $x\in\dR^{n+1}$,
$\Hess{H}(x)\geq 0$ as a quadratic form, and thus $H$ is convex on
$\dR^{n+1}$.  Unfortunately, since $\sum_{j=1}^{n+1} \pd^2_{ij} H(x)=0$ for
any $i\in\{1,\ldots,n+1\}$, the null space of $\Hess{H}$ contains $\Vo_{n+1}$
and therefore, the measure $\mu$ on $\dR^{n+1}$ defined by
$d\mu(x):=\exp\PAR{-H(x)}\,dx$ cannot be normalised into a probability measure
since $Z_\mu:=\mu(\dR^{n+1})=+\infty$.  Nevertheless, suppose that there
exists a constant $\rho>0$ such that $V''_{\{i,j\}}(u)\geq \rho$ for any
$u\in\dR$ and any $i,j\in\{1,\ldots,n\}$.  Then, $u\in\dR\mapsto
V_{\{i,j\}}(u)-\rho u^2/2$ is convex and the latter implies that
$$
\Hess{H}(x) \geq \rho\,\bI_{n+1}-(n+1)^{-1}\rho\,\Mo_{n+1},
$$
as quadratic forms. Thus, by writing
$\dR^{n+1}=\dR\Vo_{n+1}\overset{\perp}{\oplus}\cH_n$ where $\cH_n$ is the
hyperplane of equation $h_1+\cdots+h_{n+1}=0$, we get that the spectrum of
$\Hess{H}(x)$ is of the form
$$
\{0=\la_1(x)<\la_2(x)\leq\cdots\leq\la_{n+1}(x)\}
$$
with $\la_2(x)\geq n(n+1)^{-1}\rho$.  Hence, one can define the probability
measure $\si_M$ on $\dR^n$ as in \eqref{eq:defnum} for any $M$ in $\dR$.
Moreover $\si_M$ is uniformly log-concave with a constant $n(n+1)^{-1}\rho$ and
therefore the conditional measure $\mu_M$ can be defined from $\si_M$ as a
probability measure by equation \eqref{eq:mum2num}, despite the fact that
$\mu$ is not a probability measure on $\dR^{n+1}$.  The particular case
$V_{\{i,j\}}=V$ with $V$ even and uniformly convex is considered for example
in \cite{malrieu-projection}, in terms of the associated S.D.E., in order to
study the granular media equation.

As we seen, when $H$ is uniformly strictly convex with a constant
$\rho>0$, the hypotheses of Proposition \ref{pr:num-mum} are
full-filled and hence, inequalities \eqref{in:ts-l2} and
\eqref{in:ls-l2} hold. It is quite natural to ask if \eqref{in:ts-l2}
and \eqref{in:ls-l2} remains true for symmetric but non convex
Hamiltonians $H$.  In this direction, the Bakry-Émery criterion allows
the following perturbative statement due to Ivan Gentil.  The proof,
prototype of which can be found in \cite{ledoux-lsrev}, is taken from
\cite{hel-bod} and is postponed to section \ref{se:pr:pr:ivan}.


\begin{prop}[Perturbative result]\label{pr:ivan}
  Let $H(x)=V(x_1)+\cdots+V(x_{n+1})$ with
  $$
  V(u)=\frac{u^2}{2}+F(u)
  $$
  where $F:\dR\to\dR$, and let $\si_M$ be the probability measure on
  $\dR^n$ defined by \eqref{eq:defnum}, namely
  $$
  \si_M(dx_1,\ldots,dx_n)
  =(Z_{\si_M})^{-1}
  \int_{\dR^{n}}\!\exp\BPAR{-\sum_{i=1}^n V(x_i)-V(\MSX)}\,dx_1\cdots dx_n.
  $$
  Then, for \NI{F} small enough, there exists a positive constant $\cP$
  depending only on \NI{F} such that for any $n$, any $M$ and any smooth
  $f:\dR^n\to\dR$,
  \begin{equation}\label{in:ivan}
  \var{\si_M}{f} \leq \cP\,\moy{\si_M}{\ABS{\GR f}^2}.
  \end{equation}
\end{prop}
Proposition \ref{pr:ivan} remains valid if we replace, in the definition of
$\si_M$, the square function $u\mapsto u^2/2$ by a smooth convex function
$u\mapsto \Phi(u)$, provided that there exists real constants $\al$ and $\be$
such that $0<\al\leq\be\leq 2\al$ and $\al\leq \Phi''(u) \leq \be$ for every
$u\in\dR$.  The constant $\cP$ becomes in this case
$e^{2\,\OSC{F}}/\PAR{2\al\,e^{-2\,\OSC{F}}-\be}$ for
$\OSC{F}<\log\sqrt{2\al/\be}$.

The exchangeability of the underlying measure $\mu_M$ indicates that the
perturbative approach by mean of Helffer's method which sees $\si_M$ as a
quasi-product measure with small interactions is not relevant : any reduction
of $F$ in the interaction term 
$$
V\BPAR{\MSX}
$$
affects the product term $\sum_{i=1}^n V(x_i)$. Helffer's method was
essentially developed for spins systems with \emph{boundary conditions} for
which the measure is not exchangeable. For our measure $\si_M$, one can expect
in contrast that the symmetries of $H_M$ induces a stronger result, as for
many mean field models.  In this direction, Landim, Panizo and Yau have
recently established in \cite{landim-panizo-yau} that $\mu_M$ satisfies
inequalities \eqref{in:ts-l2} and \eqref{in:ls-l2} when $H$ is of the form
$H(x)=V(x_1)+\cdots+V(x_{n+1})$ where $V(u)=u^2/2+F(u)$ with $F$ and $F'$
bounded and Lipschitz.  A simple example is given by $F(x)=P(\sin(Q(x)))$
where $P$ and $Q$ are fixed polynomials in $\dR[X]$. Their proof relies on
Lu-Yau's markovian decomposition \cite{lu-yau} and on Local Central Limit
Theorem estimates \cite{kipnis-landim}. 

Following closely \cite{landim-panizo-yau}, we are actually able to
show that measure $\si_M$ itself satisfies to \eqref{in:def-ts} and
\eqref{in:def-ls} with a constants which does not depend on $n$ and
$M$, as stated in our main result, which follows.

\begin{thm}\label{th:num-quad+bd}
  Let $H(x)=V(x_1)+\cdots+V(x_{n+1})$ with $V(u)=u^2/2+F(u)$ and let $\si_M$
  be the probability measure on $\dR^n$ defined by \eqref{eq:defnum}, namely
  $$
  \si_M(dx_1,\ldots,dx_n) =(Z_{\si_M})^{-1}
  \int_{\dR^{n}}\!\exp\BPAR{-\sum_{i=1}^n V(x_i)-V(\MSX)}\,dx_1\cdots dx_n.
  $$
  Then, if $F$ is bounded and Lipschitz, there exists a positive constant
  $\cP$ depending only on \NI{F} and \NI{F'} such that for any $n$ and $M$ and
  any smooth $f:\dR^n\to\dR$,
  \begin{equation}\label{in:nu-quad+bd-ts}
  \var{\si_M}{f} \leq \cP\,\moy{\si_M}{\ABS{\GR f}^2}.
  \end{equation}  
  Moreover, if $F''$ is also bounded, there exists a positive constant
  $\cL$ depending only on \NI{F}, \NI{F'} and \NI{F''} such that for
  any $n$ and $M$ and any smooth $f:\dR^n\to\dR$,
  \begin{equation}\label{in:nu-quad+bd-ls}
  \ent{\si_M}{f^2} \leq \cL\, \moy{\si_M}{\ABS{\GR f}^2}.
  \end{equation}  
\end{thm}

As a Corollary, we recover from Proposition \ref{pr:num-mum} and
\eqref{in:compdirf} the $L^2$ factor for the Kawasaki dynamics (cf.
\eqref{in:ts-l2} and \eqref{in:ls-l2}) obtained by
\cite{landim-panizo-yau}.

The rest of the paper is divided as follows.  The first section gives
the proof of Proposition \ref{pr:ivan}, which relies only on the
Bakry-Émery criterion.  In Section \ref{se:prelims}, we give some
preliminaries to the proof of Theorem \ref{th:num-quad+bd}. Lemma
\ref{le:cov-estims} gives some covariance bounds taken from
\cite{landim-panizo-yau}. This Lemma allows us to derive the ``one
spin Lemma'' \ref{le:onespin} by a simple application of the
Bakry-Émery criterion.  Section \ref{se:qd+bd:ip} is devoted to the
derivation of the Poincaré inequality \eqref{in:nu-quad+bd-ts} and
section \ref{se:qd+bd:ls} to the derivation of the logarithmic Sobolev
counterpart \eqref{in:nu-quad+bd-ts}.  The proofs make heavy use of
the LCLT based estimates of \cite{landim-panizo-yau} throughout Lemmas
\ref{le:cvg} and \ref{le:ceg}, but our induction in $n$ is quite
different.

It is quite natural to ask if Theorem \ref{th:num-quad+bd} remains
valid if we replace the quadratic potential $u^2/2$ by a uniformly
strictly convex potential $\Phi$. We believe that this is true, at
least for $\Phi$ such that $0<\al \leq \Phi''(u)\leq \be$.  In an
other direction, one can ask if our method remains valid for discrete
spins systems similar to those presented in \cite{martinelli-99}. It
is not clear at all for us.

%
\section{Proof of Proposition \ref{pr:ivan}}\label{se:pr:pr:ivan}
%

We give here a proof of Proposition \ref{pr:ivan}, which relies only on the
Bakry-Émery criterion. Let $\si_M^*$ the probability measure on $\dR^n$
defined by
$$
(Z_{\si_M^*})^{-1}\,\exp\BPAR{-\sum_{i=1}^n
  V(x)-\frac{1}{2}\,\BPAR{\MSX}^2\,}\,dx_1\cdots dx_n.
$$
If $\si_M^*$ satisfies a Poincaré inequality with a constant $c>0$,
then $\si_M$ satisfies a Poincaré inequality with a constant
$c\exp\PAR{2\,\OSC{F}}$.  Now, for any smooth function
$f:\dR^n\to\dR$,
$$
\moy{\si_M^*}{(\GI f)^2}=\sum_{i,j=1}^n\moy{\si_M^*}{\ABS{\pd^2_{ij} f}^2}
+\bmoy{\si_M^*}{\sum_{i=1}^n\PAR{1+F''(x_i)}\,\ABS{\pd_i f}^2}
+\bmoy{\si_M^*}{\BPAR{\sum_{i=1}^n \pd_i f}^2}.
$$
In another hand, for any $i\in\{1,\ldots,n\}$ and any
$x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n$, the Bakry-Émery criterion gives that
the one dimensional probability measure
$$
\rho_i(dx_i):=(Z_{\rho_i})^{-1}\,
\exp\BPAR{-V(x_i)-\frac{1}{2}\BPAR{\MSX}^2\,}\,dx_i
$$
satisfies a Poincaré inequality with a constant $(1/2)\,\exp(2\,\OSC{F})$,
hence, by the Bakry-Émery criterion applied reversely, we get for any smooth
function $f:\dR^n\to\dR$, by summing over $i$
$$
\sum_{i=1}^n\moy{\rho_i}{\ABS{\pd^2_{ii} f}^2} +
\sum_{i=1}^n\moy{\rho_i}{\PAR{2+F''(x_i)}\,\ABS{\pd_i f}^2} \geq
2\,e^{-2\,\OSC{F}}\sum_{i=1}^n\moy{\rho_i}{\ABS{\pd_i f}^2}.
$$
Notice that
$\rho_i=\LAW{\si_M^*}{x_i\,\vert\,x_1,\ldots,x_{i-1},x_{i+1},\ldots,x_n}$.
Therefore, by taking the expectation with respect to $\si_M^*$, we get
\begin{align*}
  \moy{\si_M^*}{(\GI f)^2} 
  &\geq \PAR{2\,e^{-2\,\OSC{F}}-1}\,\sum_{i=1}^n\moy{\si_M^*}{\ABS{\pd_i f}^2}\\
  &=: \PAR{2\,e^{-2\,\OSC{F}}-1}\,\moy{\si_M^*}{\ABS{\GR f}^2}.
\end{align*}
Thus, for $\OSC{F}$ sufficiently small ($<\log\sqrt{2}$), one can take
$$
\cP=\frac{e^{2\,\OSC{F}}}{2\,e^{-2\,\OSC{F}}-1},
$$
which is optimal when $F\equiv0$ (pure Gaussian case).

%
\section{Preliminaries to the proof of Theorem \ref{th:num-quad+bd}}
\label{se:prelims}
%

Let $\ga_{n,M}$ the Gaussian measure of mean $M/(n+1)$ and covariance matrix
$\PAR{\bI_n+\Mo_n}^{-1}$. If $B(x):=\sum_{i=1}^n
F(x_i)+F\BPAR{M-x_1-\cdots-x_n}$, on can write
$$
d\si_M(x_1,\ldots,x_n)
=(Z_{n,M})^{-1}\,\exp\PAR{-B(x)}\,d\ga_{n,M}(x_1,\ldots,x_n).
$$
Thus, $\si_M$ is a bounded perturbation of $\ga_{n,M}$, which is
log-concave with a constant $\rho$ equal to $1$, and therefore,
$\si_M$ satisfies to Poincaré and logarithmic Sobolev inequalities
with constants depending only on \NI{B} (i.e. $\NI{F}$ and $n$). Our
goal is to show that the dependence in $n$ can be dropped by taking
into account \NI{F'} and \NI{F''}. The presence of the bounded part
$F$ in $V$ and the non-product nature of $\si_M$ does not allow any
direct approach based on the Bakry-Émery criterion.

Observe that $\cov{\ga_{n,M}}{x_1}{x_2}=-(n+1)^{-1}$, and we can then expect
the same decrease in $n$ for $\cov{\si_M}{V'(x_1)}{V'(x_2)}$. This is actually
the case, as stated in the following Lemma. Notice that since $\si_M$ is
exchangeable and since $\MSX$ and $x_i$ have the same law under $\si_M$, we
have $\var{\si_M}{x_1}=-n\cov{\si_M}{x_1}{x_2}$, as for $\ga_{n,M}$.


\begin{lem}\label{le:cov-estims}
  Let $\si_M$ be the probability measure on $\dR^n$ ($n>2$) defined in Theorem
  \ref{th:num-quad+bd} and $\mu_M$ the associated conditional measure defined
  by \eqref{eq:defmum}. Assume that $F$ and $F'$ are bounded, then there
  exists a constant $C>0$ depending only on \NI{F} and \NI{F'} such that for
  any $M\in\dR$
  \begin{equation}
    \label{in:cov-estim1}
    \ABS{\cov{\si_M}{V'(x_1)}{V'(x_2)}} =
    \ABS{\cov{\mu_M}{V'(x_1)}{V'(x_2)}} \leq \frac{C}{n},
  \end{equation}
  and for any $\al\in\dR$
  \begin{equation}\label{in:cov-estim2}
  \begin{split}
    \var{\si_M}{\sum_{i=1}^n V'(x_i)+V'(\MSX)}
    &=\var{\si_M}{\sum_{i=1}^n F'(x_i)+F'(\MSX)}\\
    &=\var{\mu_M}{\sum_{i=1}^{n+1} F'(x_i)} \leq nC.
  \end{split}
  \end{equation}
\end{lem}

\begin{proof}
  Inequality \eqref{in:cov-estim2} follows from \eqref{eq:mum2num} and
  \cite[Corollary 5.4]{landim-panizo-yau}.
  For \eqref{in:cov-estim1}, just write
  $$
  \cov{\si_M}{V'(x_1)}{V'(x_2)}
  =\cov{\si_M}{x_1}{x_2}
  +2\,\cov{\si_M}{x_1}{F'(x_2)}
  +\cov{\si_M}{F'(x_1)}{F'(x_2)},
  $$
  and use \eqref{eq:mum2num} and \cite[Corollary 5.3]{landim-panizo-yau} to
  estimate each term.  Actually, one can derive the estimates of
  $\cov{\si_M}{x_1}{F'(x_2)}$ and $\cov{\si_M}{x_1}{x_2}$ directly by using
  the symmetries of $\si_M$.
\end{proof}

Inequality \eqref{in:cov-estim1} of Lemma \ref{le:cov-estims} allows
us to establish the following one spin result, which is the first step
in our proof of Poincaré and logarithmic Sobolev inequalities for
$\si_M$ by induction on $n$ by mean of the Lu-Yau Markovian
decomposition. In the other hand, inequality \eqref{in:cov-estim2}
will be usefull, as we will see in sections \ref{se:qd+bd:ip} and
\ref{se:qd+bd:ls}, for the induction itself.

\begin{lem}[One spin Lemma]\label{le:onespin}
  Let $\si_M$ be the probability measure on $\dR^n$ defined in Theorem
  \ref{th:num-quad+bd}. If $F$ is bounded and Lipschitz, there exists a
  constant $A>0$ depending only on \NI{F} and \NI{F'} and not on $n$ and $M$
  such that for any $n$ and $M$ and any smooth $f:\dR\to\dR$,
  $$
  \ent{\si_M}{f(x_1)^2} \leq 2A\,\moy{\si_M}{f'(x_1)^2},
  $$
  and
  $$
  \var{\si_M}{f(x_1)} \leq A\,\moy{\si_M}{f'(x_1)^2}.
  $$
\end{lem}

\begin{proof}[Proof of Lemma \ref{le:onespin}]\label{le:pr:onespin}
  As we already noticed, it is clear that the desired inequalities are true
  with a constant depending on $n$ and $\NI{F}$, so we just have to see what
  happens for large values of $n$. We have in mind the use of the Bakry-Émery
  criterion. The Hamiltonian of the probability measure in $x_1$ is given by
  \begin{multline*}
    \vphi_{M,n}(x_1):=V(x_1)+\log Z_{M,n} %\\
    - \log \int\!\exp\PAR{-\sum_{i=2}^n V(x_i)-V(\MSX)}\, dx_2\cdots dx_n.
  \end{multline*}
  We first observe that we can forget the $F(x_1)$ part in $V(x_1)$, which is
  payed by a factor $\exp(2\,\OSC{F})$ in $A$. Hence, we simply have, after an
  integration by parts
  \begin{equation*}
    \vphi_{M,n}''(x_1)=1-\cov{\si_{M-x_1}(dx_2,\ldots,dx_n)}{V'(x_2)}{V'(x_3)}.
  \end{equation*}
  Now, \eqref{in:cov-estim1} gives $\vphi_{M,n}''(x_1) \geq 1-Cn^{-1}$, where
  $C$ is a positive constant depending only on \NI{F} and \NI{F'} and not on
  $n$ and $M$. Thus, we are able to apply the Bakry-Émery criterion for large
  values of $n$. Hence, the proof is completed, with a constant $A$ depending
  only on \NI{F} and \NI{F'} and not on $M$ and $n$.
\end{proof}

Obviously, one can replace $x_1$ in $f$ and $f'$ by $M-x_1-\cdots-x_n$ or by
any $x_i$ for $i\in\{1,\ldots,n\}$. Moreover, according to
\eqref{eq:mumeqnum}, on can replace $\moyf{\si_M}$ by $\moyf{\mu_M}$.

%
\section{Derivation of the Poincaré inequality}\label{se:qd+bd:ip}
%

This section is devoted to the derivation of inequality
\eqref{in:nu-quad+bd-ts} of Theorem \ref{th:num-quad+bd}. The proof
relies on the one spin Lemma \ref{le:onespin} and on the crucial Lemma
\ref{le:cvg} which allows us to use the Lu-Yau Markovian
decomposition.

\begin{proof}[Proof of \eqref{in:nu-quad+bd-ts}]\label{th:pr:nu-quad+bd:ts} 
  As we already noticed, the result is true with a constant depending
  on $n$, so that if we denote by $\cP_n$ the maximum of best Poincaré
  constants in dimension less than or equal to $n$, we just have to
  show that the non decreasing sequence of constants $(\cP_n)_{n\geq
    1}$ is bounded.
  
  Let us denote by $\si$ the measure $\si_M$ and by $\sik{k}$ the measure
  $\si_M$ given $x_1,\ldots,x_k$ for $k\in\{0,\ldots,n\}$ and by $f_k$ the
  conditional expectation
  $$
  \moy{\si}{f\vert x_1,\ldots,x_k}=\moy{\sik{k}}{f}.
  $$
  Notice that $\sik{k}$ is nothing else but
  $\si_{M-x_1-\cdots-x_k}(dx_{k+1},\ldots,dx_n)$. Moreover, $f_n=f$ and by
  convention $\si^{(0)}:=\si$ and thus $f_0=\moy{\mu}{f}$.  For a \emph{fixed}
  function $f$, we can always choose the order of the coordinates
  $x_1,\ldots,x_n$ such that $\moy{\si}{\ABS{\pd_k f}^2}$ becomes a \emph{non
    increasing} sequence in $k\in\{1,\ldots,n\}$. This gives
  $$
  \sum_{i=k+1}^n \frac{1}{n-k}\,\moy{\si}{\ABS{\pd_i f}^2} \leq
  \moy{\si}{\ABS{\pd_{k+1} f}^2}.
  $$
  Following Lu-Yau \cite{lu-yau}, we have the following Markovian
  decomposition of the variance
  $$
  \var{\si}{f} := \moy{\si}{f^2}-\moy{\si}{f}^2
  =\sum_{k=1}^n\moy{\si}{(f_k)^2-(f_{k-1})^2}
  =\sum_{k=1}^n\moy{\si}{\var{\sik{k-1}}{f_k}}.
  $$
  Since measure $\sik{k-1}$ integrates coordinates $x_k,\ldots,x_n$ and
  function $f_k$ depends only on coordinates $x_1,\ldots,x_k$, the quantity
  $\var{\sik{k-1}}{f_k}$ is actually a variance for a one spin function.
  Therefore, by the one spin Lemma \ref{le:onespin}, there exists a constant
  $A>0$ depending on \NI{F} and \NI{F'} but not on $n$ and $M$ such that
  $$
  \var{\si}{f} \leq A\,\sum_{k=1}^n \moy{\si}{\ABS{\pd_k f_k}^2}.
  $$
  Our aim is to express the right hand side of the previous inequality in
  terms of $\ABS{\pd_k f}^2$. Notice that the $k=n$ term in the sum is trivial
  since $f_n=f$. By definition of $f_k$, we get for any $k\in\{1,\ldots,n-1\}$
  $$
  \pd_k f_k=\moy{\sik{k}}{\pd_k f}-\bcov{\sik{k}}{f}{V'(\MSX)}.
  $$
  At this stage, we notice that by $n-k$ integrations by parts, we have
  $$
  \bcov{\sik{k}}{f}{V'(\MSX)}
  =\frac{1}{n-k}\, \sum_{i=k+1}^n\cov{\sik{k}}{f}{V'(x_i)}
  -\frac{1}{n-k}\,\sum_{i=k+1}^n \moy{\sik{k}}{\pd_i f}.
  $$
  Therefore, we can write by denoting $S_k:=\sum_{i=k+1}^n V'(x_i)+V'(\MSX)$
  $$
  \pd_k f_k =\moy{\sik{k}}{\pd_k f} -\frac{1}{n-k+1}\,\cov{\sik{k}}{f}{S_k}
  +\frac{1}{n-k+1}\,\sum_{i=k+1}^n \moy{\sik{k}}{\pd_i f}.
  $$
  Now, by the Cauchy-Schwarz inequality
  $$
  \ABS{\pd_k f_k}^2 
  \leq 3\,\moy{\sik{k}}{\ABS{\pd_k f}^2}
  +\frac{3}{(n-k)^2}\,\cov{\sik{k}}{f}{S_k}^2
  +\frac{3}{n-k}\,\sum_{i=k+1}^n\moy{\sik{k}}{\ABS{\pd_i f}^2}.
  $$
  This gives by summing over all $k$ in $\{1,\ldots,n-1\}$ (the case $k=n$
  is trivial)
  \begin{align*}
    \sum_{k=1}^{n-1}\moy{\si}{\ABS{\pd_k f_k}^2}
    \leq\;&3\,\moy{\si}{\ABS{\GR f}^2}
          +3\,\sum_{k=1}^{n-1}
          \frac{1}{(n-k)^2}\,\moy{\si}{\cov{\sik{k}}{f}{S_k}^2}\\
          &+3\,\sum_{k=1}^{n-1}
          \frac{1}{n-k}\,\sum_{i=k+1}^n\moy{\si}{\ABS{\pd_if}^2}.
  \end{align*}
  The monotonicity of $\moy{\si}{\ABS{\pd_i f}^2}$ yields
  $$
  \sum_{k=1}^{n-1}
  \moy{\si}{\ABS{\pd_k f_k}^2} \leq\;6\,\moy{\si}{\ABS{\GR f}^2}
  +3\,\sum_{k=1}^{n-1}\frac{1}{(n-k)^2}\,\moy{\si}{\cov{\sik{k}}{f}{S_k}^2}.
  $$
  By inequality \eqref{in:cvg} of Lemma \ref{le:cvg}, there exists a
  positive constant $C$ depending only on \NI{F} and \NI{F'} such that for any
  $\veps>0$, there exists a positive constant $C_\veps$ depending only on
  \NI{F}, \NI{F'} and $\veps$ such that for any $k\in\BRA{1,\ldots,n-1}$
  $$
  \cov{\sik{k}}{f}{S_k}^2 \leq (C_\veps+\veps (n-k)C)\,\var{\sik{k}}{f}
  +(n-k)C_\veps\,\sum_{i=k+1}^n\moy{\sik{k}}{\ABS{\pd_i f}^2}.
  $$
  Therefore, by the monotonicity of $\moy{\si}{\ABS{\pd_i f}^2}$
  again
  \begin{equation}\label{in:pr:estim:grad-grad-var-var}
  \begin{split}
    \sum_{k=1}^{n-1}\moy{\si}{\ABS{\pd_k f_k}^2}
    \leq\;&C'_\veps\,\moy{\si}{\ABS{\GR f}^2}
    +C'_\veps\,\sum_{k=1}^{n-1}%
    \frac{1}{(n-k)^2}\,\moy{\si}{\var{\sik{k}}{f}} \\
    &+\veps
    C'\,\sum_{k=1}^{n-1}\frac{1}{n-k}\,\moy{\si}{\var{\sik{k}}{f}}.
  \end{split}
  \end{equation}
  Recall that $\cP_n$ is the maximum of best Poincaré constants in dimension
  less than or equal to $n$. The last sum of the right hand side (RHS) of
  \eqref{in:pr:estim:grad-grad-var-var} can be bounded above as follows
  $$
  \sum_{k=1}^{n-1}\frac{1}{n-k}\moy{\si}{\var{\sik{k}}{f}} 
  \leq \cP_{n-1} \moy{\si}{\ABS{\GR f}^2}.
  $$
  It remains to examine the first sum of the RHS of
  \eqref{in:pr:estim:grad-grad-var-var}.  The Jensen inequality yields
  $$
  \moy{\si}{\var{\sik{k}}{f}} \leq \var{\si}{f},
  $$
  and therefore, we get for any $p\in\{1,\ldots,n-1\}$
  \begin{align*}
    \sum_{k=1}^{n-1}\frac{1}{(n-k)^2}\,\moy{\si}{\var{\sik{k}}{f}}
    &=\var{\si}{f}\sum_{k=1}^{n-p-1}\frac{1}{(n-k)^2}
    +\sum_{k=n-p}^{n-1}\frac{1}{(n-k)^2}\,\moy{\si}{\var{\sik{k}}{f}}\\
    &\leq\var{\si}{f}\sum_{k=p+1}^{n-1}\frac{1}{k^2}
    +\sum_{k=1}^{p}\frac{1}{k^2}\,\moy{\si}{\var{\sik{n-k}}{f}}.
  \end{align*}
  At this stage, we observe that for every $k$ in $\{1,\ldots,p\}$,
  $$
  \moy{\si}{\var{\sik{n-k}}{f}} 
  \leq \cP_{p} \sum_{i=n-k+1}^n \moy{\si}{\ABS{\pd_i f}^2}
  \leq p\cP_{p}\,\moy{\si}{\ABS{\pd_{n-p+1} f}^2}.
  $$
  We are now able to collect our estimates of the RHS of
  \eqref{in:pr:estim:grad-grad-var-var}. Putting all together, we have
  obtained that
  $$
  \sum_{k=1}^{n-1}\moy{\si}{\ABS{\pd_k f_k}^2}
  \leq 
  (C'_\veps+p\pi^2\cP_{p}C'_\veps+\veps C'\cP_{n-1})\,\moy{\si}{\ABS{\GR f}^2}
  +(C'_\veps R_p)\var{\si}{f},
  $$
  where $R_p:=\sum_{k=p+1}^{n-1}k^{-2}$. Therefore, for somme
  $C''_{p,\veps}>0$,
  $$
  (1-AC'_\veps R_p)\,\var{\si}{f}
  \leq (C''_{p,\veps}+\veps AC'\cP_{n-1})\,\moy{\si}{\ABS{\GR f}^2}.
  $$
  Now, we may choose $\veps < 1/(AC')$ and then $p$ large enough (always
  possible when $n$ is sufficiently large) to ensure that
  $$
  R_p < \min\PAR{\frac{1}{AC'_\veps},\frac{1-\veps AC'}{AC'_\veps}}.
  $$
  This gives two positive constants $\al$ and $\be$ with $\be<1$ depending
  only on \NI{F} and \NI{F'} such that for large values of $n$, one has
  $\cP_n \leq \al + \be\,\cP_{n-1}$, and therefore $\sup_{n} \cP_n < +\infty$.
\end{proof}

Let us give now the crucial Lemma which allows us to use the Markovian
decomposition of Lu-Yau, by splitting the covariance term into a
variance term and a gradient term. The proof makes heavy use of
estimates taken from \cite{landim-panizo-yau}.

\begin{lem}\label{le:cvg} 
  Let $\si_M$ be the probability measure on $\dR^n$ defined in Theorem
  \ref{th:num-quad+bd}. Assume that $F$ is bounded and Lipschitz, then there
  exists a positive constant $C$ depending only on $\NI{F}$ and $\NI{F'}$ such
  that for any $\veps>0$, there exists a positive constant $C_\veps$ depending
  only on $\NI{F}$, $\NI{F'}$ and $\veps$ such that for any $n\in\dN^*$, any
  $M\in\dR$ and any smooth function $f:\dR^n\to\dR$
 \begin{equation}\label{in:cvg}   
  \cov{\si_M}{f}{S}^2
  \leq (C_\veps+\veps n C)\,\var{\si_M}{f}
  +nC_\veps\,\moy{\si_M}{\ABS{\GR f}^2},
 \end{equation}
 where $S:=\sum_{i=1}^n V'(x_i)+V'(\MSX)$.
\end{lem}

\begin{proof}[Proof of Lemma \ref{le:cvg}]\label{le:pr:cvg}
  Notice that we just have to study what happens for small values of $\veps$
  and large values of $n$, since for any $\veps>0$ and any $n\leq n_\veps$, we
  get by the Cauchy-Schwarz inequality and \eqref{in:cov-estim2} that
  $$
  \cov{\si_M}{f}{S}^2 \leq n_\veps C\,\var{\si_M}{f} =:
  C_\veps\,\var{\si_M}{f}.
  $$
  We have in mind the use of the partitioning result of
  \cite{landim-panizo-yau}. If ${\mu_M}$ denotes the conditional measure on
  $\dR^{n+1}$ associated to $\si_M$ as in \eqref{eq:mum2num}, we have
  $$
  \cov{{\si_M}}{f}{S(x_1,\ldots,x_n)}^2
  =\bcov{{\mu_M}}{f}{\sum_{i=1}^{n+1}F'(x_i)}^2.
  $$
  Now, for $n$ large enough, one can then subdivide the set
  $\{1,\ldots,n+1\}$ into $\ell$ adjacent subsets $I_i$ of size $K$ or $K+1$.
  We have in mind to take $K^{-1}\leq\veps$, which is always possible when $n$
  is large enough. We can write with this decomposition
  $$
  \bcov{{\mu_M}}{f}{\sum_{i=1}^{n+1} F'(x_i)} =
  \bcov{{\mu_M}}{f}{\sum_{i=1}^\ell\sum_{k\in I_i} F'(x_k)}.
  $$
  For any $(i,x)\in\BRA{1,\ldots,\ell}\times\dR^{n+1}$, we define the
  ``total spin on $I_i$'' by $M_i(x):=\sum_{k\in I_i} x_k$.  On $\dR^{I_i}$,
  one can define the conditional measure $\mu_{M_i}$ with total spin $M_i$,
  as in \eqref{eq:defmum}. To lighten the notations, we denote this measure by
  $\muMi$.  We get from the latter by the Cauchy-Schwarz inequality
  \begin{equation}\label{in:cvg-step-1}
    \begin{split}
      \bcov{{\mu_M}}
      {f}{\sum_{i=1}^{n+1}F'(x_i)}^2 
      \leq\;&2\,\bcov{{\mu_M}}{f}
      {\sum_{i=1}^\ell\sum_{k\in I_i}\PAR{F'(x_k)-\moy{\muMi}{F'}}}^2\\
      &+ 2\,\bcov{{\mu_M}} {f}{\sum_{i=1}^\ell\ABS{I_i}\moy{\muMi}{F'}}^2
    \end{split}
  \end{equation}
  By the Cauchy-Schwarz inequality again and by \eqref{eq:mumeqnum},
  the second term of the RHS of \eqref{in:cvg-step-1} can be bounded
  above by
  $$
  \var{\si_M}{f}\bvar{\mu_M}{\sum_{i=1}^\ell\ABS{I_i}\moy{\muMi}{F'}}.
  $$
  Now, according to \cite[ineq. (3.10)]{landim-panizo-yau}, the last
  variance in the RHS is bounded above by $nC/K$ for $n$ sufficiently large,
  which can be rewritten as $\veps nC$.  We turn now to the control of the
  first term of the RHS of \eqref{in:cvg-step-1}. Since
  $\moyf{{\mu_M}}=\moyf{{\mu_M}} \circ \moyf{\muMi}$, we get
  $$
  \bcov{{\mu_M}}
  {f}
  {\sum_{i=1}^\ell\sum_{k\in I_i}\PAR{F'(x_k)-\moy{\muMi}{F'}}}
  =\sum_{i=1}^\ell\bmoy{{\mu_M}}{\bcov{\muMi}{f}{\sum_{k\in I_i} F'(x_k)}}.
  $$
  Thus, the Cauchy-Schwarz inequality yields
  $$
  \bcov{{\mu_M}}
  {f}
  {\sum_{i=1}^\ell\sum_{k\in I_i}\PAR{F'(x_k)-\moy{\muMi}{F'}}}^2
  \leq
  \ell\,
  \sum_{i=1}^\ell\bmoy{{\mu_M}}{\bcov{\muMi}{f}{\sum_{k\in I_i} F'(x_k)}^2}.
  $$
  Again by the Cauchy-Schwarz inequality, we get
  $$
  \bcov{\muMi}{f}{\sum_{k\in I_i} F'(x_k)}^2 
  \leq
  \var{\muMi}{f}\bvar{\muMi}{\sum_{k\in I_i}F'(x_k)}.
  $$
  By virtue of \eqref{in:cov-estim2} applied to $\muMi$, we obtain
  $$
  \bcov{\muMi}{f}{\sum_{k\in I_i} F'(x_k)}^2
  \leq C\,\ABS{I_i}\,\var{\muMi}{f}.
  $$
  Now, for any $i$, let $r_i=\max\{k,\,k\in I_i\}$
  and $J_i:=I_i\bs\{r_i\}$ and $\siMi$
  the probability measure on $\dR^{J_i}$ associated with the Hamiltonian
  $$
  \sum_{k\in J_i} V(x_k)+V(M_i-\sum_{k\in J_i} x_k).
  $$
  Equation \eqref{eq:mumeqnum} simply gives
  $$
  \var{\muMi}{f} = \var{\siMi}{f(\vphi_i(x))}, 
  $$
  where $\vphi_i:\dR^{n+1}\to\dR^n$ is defined by
  $$
  (\vphi_i(x))_k
  :=
  \begin{cases}
    x_k & \text{if $k\neq r_i$}\\
    M_i-\sum_{l\in J_i} x_l & \text{if $k=r_i$}
  \end{cases}
  $$
  Recall that $\cP_K$ is the maximum of the best Poincaré constants for
  $\si_M$ in dimensions less than or equal to $K$. We get by definition of
  $\cP_K$ that
  \begin{align*}
    \var{\siMi}{f} 
    &\leq\cP_K\sum_{k\in J_i}
    \moy{\siMi}{\ABS{(\pd_k f)(\vphi_i)-(\pd_{r_i}f)(\vphi_i)}^2}\\
    &=\cP_K\sum_{k\in J_i}
    \moy{\muMi}{\ABS{\pd_k f-\pd_{r_i}f}^2}.
  \end{align*}
  Hence, by the Cauchy-Schwarz inequality, we get
  $$
  \var{\siMi}{f}
  \leq
  2\cP_K\,\bmoy{\muMi}{\sum_{k\in J_i} \ABS{\pd_k f}^2} +
  2(\ABS{I_i}-1)\cP_K\,\moy{\muMi}{\ABS{\pd_{r_i} f}^2},
  $$
  Summarising, since $\cP_K$ depends only on $K$, \NI{F}, \NI{F'},
  we have obtained that the first term of the right hand side of
  \eqref{in:cvg-step-1} is bounded above by
  $$
  nC_K\cP_K\,\moy{\mu_M}{\ABS{\GR f}^2},
  $$
  which can be rewritten by virtue of \eqref{eq:mumeqnum} as
  $n\,C'_\veps\,\moy{{\si_M}}{\ABS{\GR f}^2}$. This concludes the proof of
  \eqref{in:cvg} and Lemma \ref{le:cvg}.
\end{proof}

%
\section{Derivation of the Logarithmic Sobolev inequality}\label{se:qd+bd:ls}
%

This section is devoted to the derivation of inequality
\eqref{in:nu-quad+bd-ls} of Theorem \ref{th:num-quad+bd}. As for the
Poincaré inequality \eqref{in:nu-quad+bd-ts}, the proof relies on the
one spin Lemma \ref{le:onespin} and on a crucial Lemma \ref{le:ceg}
which allows us to use the Lu-Yau Markovian decomposition.

\begin{proof}[Proof of the logarithmic Sobolev inequality
  \eqref{in:nu-quad+bd-ls} of Theorem
  \ref{th:num-quad+bd}]\label{th:pr:nu-quad+bd:ls}
  
  We follow here the same scheme used for the Poincaré inequality.  For any
  smooth non negative function $g:\dR^n\to\dR^+$, we have the
  following decomposition of the entropy
  \begin{align*}
    \ent{\si}{g} :&= \moy{\si}{g\log g}-\moy{\si}{g}\log\moy{\si}{g}\\
    &=\sum_{k=1}^n \moy{\si}{g_k\log g_k-g_{k-1}\log g_{k-1}}\\
    &=\sum_{k=1}^n \moy{\si}{\ent{\sik{k-1}}{g_k}}.
  \end{align*}
  Alike for the variance, measure $\sik{k-1}$ integrates on
  $x_k,\ldots,x_n$ and function $f_k$ depends only on
  $x_1,\ldots,x_k$, so that $\ent{\sik{k-1}}{g_k}$ is actually an
  entropy for a one spin function.  Therefore, by the one spin Lemma
  \ref{le:onespin}, there exists a positive constant $A$ depending on
  \NI{F} and \NI{F'} but not on $n$ and $M$ such that
  $$
  \ent{\si}{g}
  \leq 2A\,\sum_{k=1}^n \bmoy{\si}{\frac{\ABS{\pd_k g_k}^2}{4g_k}},
  $$
  By taking $g=f^2$ for a smooth function $f:\dR^n\to\dR$, we get
  $$
  \ent{\si}{f^2}
  \leq 2A\,\sum_{k=1}^n \bmoy{\si}{\frac{\ABS{\pd_k (f^2)_k}^2}{4(f^2)_k}}.
  $$
  By imitating the method used for the Poincaré inequality, we get
  that
  $$
  \frac{\ABS{\pd_k (f^2)_k}^2}{4(f^2)_k}
  \leq 3\,\frac{\ABS{\moy{\sik{k}}{f\pd_k f}}^2}{\moy{\sik{k}}{f^2}}
  +\frac{6}{(n-k)^2}\,\frac{\cov{\sik{k}}{f^2}{S_k}^2}{\moy{\sik{k}}{f^2}}
  +\frac{3}{n-k}\,\sum_{i=k+1}^n 
  \frac{\ABS{\moy{\sik{k}}{f\pd_i f}}^2}{\moy{\sik{k}}{f^2}}.
  $$
  The Cauchy-Schwarz inequality yields
  $$
  \frac{\ABS{\moy{\sik{k}}{f\pd_k f}}^2}{\moy{\sik{k}}{f^2}}
  \leq\moy{\sik{k}}{\ABS{\pd_k f}^2}.
  $$
  Therefore, the Jensen inequality and the monotonicity of
  $\moy{\si}{\ABS{\pd_i f}^2}$ yield
  $$
  \sum_{k=1}^{n-1}\moy{\si}{\frac{\ABS{\pd_k (f^2)_k}^2}{4(f^2)_k}}
  \leq\; 6\,\moy{\si}{\ABS{\GR f}^2}
  +6\,\sum_{k=1}^{n-1}\frac{1}{(n-k)^2}\,
  \moy{\si}{\frac{\cov{\sik{k}}{f^2}{S_k}^2}{\moy{\sik{k}}{f^2}}}.
  $$
  By inequality \eqref{in:ceg} of Lemma \eqref{le:ceg}, there
  exists a positive constant $C$ depending only on \NI{F}, \NI{F'} and
  \NI{F''} such that for any $\veps>0$, there exists a positive
  constant $C_\veps$ depending only on \NI{F}, \NI{F'}, \NI{F''} and
  $\veps$ such that for any $n$ and $M$
  $$
  \frac{\cov{\sik{k}}{f^2}{S_k}^2}{\moy{\sik{k}}{f^2}}
  \leq (C_\veps+\veps(n-k)C)\,\ent{\sik{k}}{f^2}
  +(n-k)C_\veps\,\sum_{i=k+1}^n\moy{\sik{k}}{\ABS{\pd_i f}^2}.
  $$
  Hence, we are now able to proceed as the same way as for the Poincaré
  inequality.
\end{proof}

As for the derivation of the Poincaré inequality, we give now the
crucial Lemma which allows us to use the Markovian decomposition of
Lu-Yau.

\begin{lem}\label{le:ceg} 
  Let $\si_M$ be the probability measure on $\dR^n$ defined in Theorem
  \ref{th:num-quad+bd}. Assume that $F$, $F'$ and $F''$ are bounded, then
  there exists a positive constant $C$ depending only on $\NI{F}$, $\NI{F'}$
  and $\NI{F''}$ such that for any $\veps>0$, there exists a positive constant
  $C_\veps$ depending only on $\NI{F}$, $\NI{F'}$, $\NI{F''}$ and $\veps$ such
  that for any $n\in\dN^*$, any $M\in\dR$ and any smooth function
  $f:\dR^n\to\dR$ such that $\moy{\si_M}{f^2}=1$
  \begin{equation}\label{in:ceg}   
    \cov{\si_M}{f^2}{S}^2
    \leq (C_\veps+\veps n C)\,\ent{\si_M}{f^2}
    +nC_\veps\,\moy{\si_M}{\ABS{\GR f}^2},
  \end{equation}
  where $S(x):=\sum_{i=1}^n V'(x_i)+V'(\MSX)$.
\end{lem}

\begin{proof}[Proof of Lemma \ref{le:ceg}]\label{le:pr:ceg}
  We follow the same scheme as for \eqref{in:cvg}, by replacing the
  Cauchy-Schwarz inequality by the entropy inequality.  Since $f^2$ is
  a density with respect to $\si_M$, we can write
  $$
  \cov{\si_M}{f^2}{S}=\moy{\si_M}{\PAR{S-\moy{\si_M}{S}}\,f^2},
  $$
  and hence, we get by the entropy inequality that for any $\be>0$
  $$
  \cov{\si_M}{f^2}{S} \leq
  \be^{-1}\,\log\moy{\si_M}{\exp\PAR{\be\PAR{S-\moy{\si_M}{S}}}}
  +\be^{-1}\,\moy{\si_M}{f^2\log f^2}.
  $$
  By \eqref{eq:mum2num} and \cite[Lemma 6.1]{landim-panizo-yau},
  the first term of the right hand side is bounded above by $nC\be$
  where $C$ depends only on \NI{F}, and \NI{F''}. This yields by
  considering the minimum in $\be>0$
  $$
  \cov{\si_M}{f^2}{S}^2\leq nC\,\moy{\si_M}{f^2\log f^2}.
  $$
  Thus, for any fixed $\veps>0$, we just have to study what happens
  for large values of $n$ since $nC\leq n_\veps C=:C_\veps$ for $n\leq
  n_\veps$.  After rewritting \eqref{in:ceg} in terms of ${\mu_M}$, we
  get by Cauchy-Schwarz's inequality
  \begin{equation}\label{in:ceg-step-1}
    \begin{split}
    \bcov{{\mu_M}}
    {f^2}{\sum_{i=1}^{n+1}F'(x_i)}^2 
    \leq\;&
    2\,\bcov{{\mu_M}}
    {f^2}{\sum_{i=1}^\ell\sum_{k\in I_i}\PAR{F'(x_k)-\moy{\muMi}{F'}}}^2\\
    &+2\,\bcov{{\mu_M}}{f^2}{\sum_{i=1}^\ell\ABS{I_i}\moy{\muMi}{F'}}^2.
    \end{split}
  \end{equation}
  Let us treat the first term of the right hand side of
  \eqref{in:ceg-step-1}. It can be rewritten as
  $$
  2\,\sum_{i=1}^\ell
  \bmoy{\mu_M}{\moy{\muMi}{f^2}\bcov{\muMi}{f_i^2}{\sum_{k\in I_i} F'(x_k)}},
  $$
  where $f_i^2 :=f^2/\moy{\muMi}{f^2}$. Thus, by the Cauchy-Schwarz
  inequality, the first term of the RHS of \eqref{in:ceg-step-1} is
  bounded above by
  $$
  2\,\ell\,\sum_{i=1}^\ell
  \bmoy{{\mu_M}}
  {\moy{\muMi}{f^2}\bcov{\muMi}{f_i^2}{\sum_{k\in I_i} F'(x_k)}^2},
  $$
  where we used the Jensen inequality with respect to the density
  $\moy{\muMi}{f^2}$.  Now, by the entropy inequality and by
  \cite[Lemma 6.1]{landim-panizo-yau}
  $$
  \moy{\muMi}{f^2} 
  \bcov{\muMi}{f_i^2}{\sum_{k\in I_i} F'(x_k)}^2
  \leq C\,\ABS{I_i}\,\ent{\muMi}{f^2}.
  $$
  At this stage, the argument used for the Poincaré inequality can
  be rewritten exactly in the same way, by replacing the variance by
  the entropy and $\cP_K$ by $\cL_K$. It gives finally that the first
  term of the RHS of \eqref{in:ceg-step-1} is bounded above by
  $$
  nC_K\cL_K\,\moy{\mu_{M}}{\ABS{\GR f}^2}.
  $$
  The latter can be rewritten by virtue of \eqref{eq:mumeqnum} as
  $n\,C'_\veps\,\moy{{\si_M}}{\ABS{\GR f}^2}$.  It remains to bound
  the last term of the RHS of \eqref{in:ceg-step-1}.  Let $\be_0$ as
  in \cite[Lemma 6.5]{landim-panizo-yau} and $\de\in(0,2)$.  By a
  simple rewriting of \cite[Lemma 4.5]{landim-panizo-yau}, one gets
  that if $\ent{\mu_M}{f^2} \leq \de(n+1)\be_0^2$ with $n$ and $K$
  large enough
  $$
  \bcov{{\mu_M}}
  {f^2}{\sum_{i=1}^\ell\ABS{I_i}\moy{\muMi}{F'}}^2
  \leq \de n C\,\ent{\mu_M}{f^2}.
  $$
  In another hand, if $\ent{\mu_M}{f^2} \geq \de(n+1)\be_0^2$, one gets
  $$
  \bcov{{\mu_M}} {f^2}{\sum_{i=1}^\ell\ABS{I_i}\moy{\muMi}{F'}}^2 \leq \de n
  C_K\,\ent{\mu_M}{f^2} + C_{K,\de}\,n\,\moy{\mu_M}{\ABS{\GR f}^2}.
  $$
  This last estimate is based on a simple rewriting of \cite[Lemma
  4.5]{landim-panizo-yau} together with the following straightforward but
  essential version of \cite[Lemma 4.6]{landim-panizo-yau} :
  $$
  \moy{\nu_{I_i\cup I_j,M}}{(m_i-m_j)^2f^2} \leq C_1(K)\,\moy{\nu_{I_i\cup
      I_j,M}}{f^2} + C_2(K)\,\cL_{2K}\,\bmoy{\nu_{I_i\cup I_j,M}}{\sum_{k\in
      I_i\cup I_j} \ABS{\pd_k f}^2},
  $$
  where $\nu_{I_i\cup I_j,M}$ is the conditional measure on
  $I_i\cup I_j$, $m_i=\ABS{I_i}^{-1}\sum_{k\in I_i}$, and $C_1(K)\to
  0$ when $K\to+\infty$.
  
  Summarising, we get that for any $\de\in (0,2)$ and for $n$ and $K$
  large enough, the last term of the RHS of \eqref{in:ceg-step-1} is
  bounded above as follows
  $$
  \bcov{{\mu_M}} {f^2}{\sum_{i=1}^\ell\ABS{I_i}\moy{\muMi}{F'}}^2 
  \leq %
  n\de C_K\,\ent{{\mu_M}}{f^2}
  +nC_{K,\de}\cL_{2K}\,\moy{\mu_M}{\ABS{{\GR}f}^2},
  $$
  which can be rewritten by virtue of \eqref{eq:mumeqnum} as $\veps
  n C'\,\ent{{\si_M}}{f^2} + nC''_\veps\,\moy{\si_M}{\ABS{\GR f}^2}$.
  This achieves the proof of \eqref{in:ceg} and Lemma \ref{le:ceg}.
\end{proof}

%
\section*{Acknowledgments}
%

The author would like to warmly acknowledge Prof. Michel Ledoux for
helpful discussions and encouraging comments, and Doct. Ivan Gentil
for some discussions at the beginning of this work.

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\begin{center}
  \hrule
\end{center}

\noindent
\textbf{E-mail}: \url{mailto:chafai@cict.fr}.\\
\textbf{URL} : \url{http://www.lsp.ups-tlse.fr/Chafai/}\\
\textbf{Postal Adress}: 
Laboratoire de Statistique et Probabilités, U.M.R.
C.N.R.S. C5583, Université Paul Sabatier, 118 route de Narbonne, 31062
Toulouse CEDEX 4, France.\\
\textbf{Keywords}: \mykeywords.\\
\textbf{Subj. Class. MSC-2000} : \mysubjclass.

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