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Conservative spin systems, Poincare inequality, 
Ginzburg-Landau process, spectral gap.
---------------0202040814767
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\begin{document}

\title[Poincar\'e inequalities in conservative systems]
{Uniform 
Poincar\'e inequalities for unbounded conservative spin
systems: The non--interacting case}
\author{Pietro Caputo}
\address{Dip. Matematica, Universita' di Roma Tre, L.go S. Murialdo 1,
00146 Roma, Italy.}
\email{caputo\@@mat.uniroma3.it}


\begin{abstract}
We prove
a uniform Poincar\'e inequality for  
non--interacting unbounded spin systems with 
a conservation law, when the single--site potential is a 
bounded perturbation of a convex function. 
The result is then applied to Ginzburg-Landau 
processes to show diffusive scaling of the associated spectral gap.

\bigskip

\noindent
{\em 2000 MSC: 60K35}

\noindent
{\em Key words:} Conservative spin systems, Poincar\'e inequality,
Ginzburg-Landau process, spectral gap.


\end{abstract}
%\footnote{{\em 2000 MSC: 60K35}}
\date{\today}

\maketitle



\section{Introduction and main result}

Consider a probability measure $\mu$ on $\bbR$ of the form
\be
\mu(\dd \eta)=\frac{\nep{-V(\eta)}}{Z}\,\,\dd \eta\,,
\la{ein}
\end{equation}
with $Z = \int\nep{-V(\eta)}\dd \eta$. Denote by $\mu_N$ the
$N$-fold product measure obtained by tensorization
of $\mu$ on $\bbR^N$, $N\in\bbN$. The canonical 
Gibbs measure with density $\r\in\bbR$ is defined 
by conditioning $\mu_N$ on the $N-1$ dimensional hyperplane
$\sum_{i=1}^N\eta_i = \r N$, i.e.\ 
\be
\nu_{N,\r}=\mu_N\Big(\,\cdot\,\Big|\, \sum_{i=1}^N\eta_i = \r N\Big)\,.
\la{zwei}
\end{equation}
We are going to give sufficient conditions on the potential
$V$ in order that the canonical measures $\nu_{N,\r}$ satisfy a 
Poincar\'e inequality, uniformly in $\r$ and $N$. 
%The latter is defined
%as follows.
\bigskip

%\subsection{Poincar\'e inequalities}
For any probability measure $\nu$ we write 
$\nu(F) = \int F\dd\nu$ for the mean of a function $F$ and 
$\var_\nu(F)$ for the variance $\nu(F^2) - \nu(F)^2$. For any smooth 
function $F$ on $\bbR^N$ we write $\partial_iF$ for 
the partial gradient along the $i$-th coordinate.
We say that a measure $\nu$ on $\bbR^N$ satisfies a 
Poincar\'e inequality if there exists a finite constant $\g$ 
such that 
$$
\var_\nu(F) \leq \g\,\sum_{i=1}^N \nu\big[
(\partial_iF)^2\big]
$$
holds for every smooth, real function $F$. 
Specializing to the canonical Gibbs measures (\ref{zwei}) we 
define the quadratic form 
$$
\cE_{N,\r}(F) = \sum_{i=1}^N\nu_{N,\r}\big[(\partial_iF)^2\big]\,.
$$
For every $N\in\bbN$ and $\r\in\bbR$ the Poincar\'e constant is given by
\be
\g(N,\r)=\sup_{F}\frac{\var_{\nu_{N,\r}}(F)}{\cE_{N,\r}(F)}
\la{gamma}
\end{equation}
where the supremum is carried over all 
smooth, real functions $F$ on $\bbR^N$.
We say that a uniform Poincar\'e inequality holds 
whenever 
\be
\sup_{N\in\bbN}\,\sup_{\r\in\bbR}\,
\g(N,\r)\,<\,\infty\,.
\la{drei}
\end{equation}

\bigskip

%\subsection{Main result}
The main result of this paper states that such an estimate
holds when $V$ is a bounded perturbation of a strictly convex function. 
\begin{Th}
\la{teorema}
Let $V$ be a measurable function of the form 
$V=\varphi+\psi$ with $\varphi\in\cC^2(\bbR)$ and  
$$
\inf_{x\in\bbR}\varphi''(x)
%\frac{\dd^2\varphi}{\dd x^2}
\,>\,0\,,\quad\quad
\sup_{x\in\bbR}|\psi(x)|\,<\,\infty\,.
$$
Then the measures $\nu_{N,\r}$ satisfy a uniform Poincar\'e inequality.
\end{Th}
The proof of Theorem \ref{teorema} will be given in the next two sections.
It relies on a simple but powerful idea recently introduced
by Carlen, Carvalho and Loss in \cite{CCL}. A similar technique 
was then used also in \cite{CapMa} to study 
the relaxation to equilibrium for
a conservative lattice gas dynamics. The argument of \cite{CCL} 
essentially shows that in view of 
the symmetry of the measures (\ref{zwei}) one can
reduce the problem to the analysis of a one--dimensional process.
The latter will be studied by means of a local limit theorem expansion.
 

\bigskip


Poincar\'e inequalities for conservative systems are usually studied
on the level of the corresponding Ginzburg-Landau or Kawasaki dynamics,
\cite{LY,BZ1,BZ2,CM}.
This is an ergodic diffusion process on the hyperplane 
$\sum_{i=1}^N\eta_i = \r N$, with $\nu_{N,\r}$
as reversible invariant measure, and Dirichlet form of the type
$$
\cD_{N,\r}(F) = \sum_{i=1}^{N-1}\nu_{N,\r}\big[(\partial_{i+1}F -
\partial_i F)^2\big]\,.
$$
In this context the Poincar\'e inequality becomes
a statement about the gap in the spectrum of the associated 
self adjoint Markov generator, or equivalently about the 
rate of convergence to equilibrium in the $L^2(\nu_{N,\r})$--norm.
In section 4 we shall see that an immediate corollary of 
Theorem \ref{teorema} is an estimate of the form
\be
\var_{\nu_{N,\r}}(F) \leq k\,N^2\,\cD_{N,\r}(F) 
\la{vier}
\end{equation}
for all smooth functions $F$ with 
a constant $k$ independent of the density $\r$. 
This says that the spectral gap scales diffusively with the
size of the system, uniformly in the density. Such estimates 
are usually a key step in establishing hydrodynamical limits,
see \cite{KL}. The question of the generality under which 
estimate (\ref{vier}) holds was already raised in \cite{V}. 
It was pointed out that when $V$ is a strictly convex 
function then (\ref{vier}) holds. Indeed, in this case 
a general argument based on the Bakry--Emery criterium applies,
see \cite{Cap,Djalil}. More directly, when there is no perturbation ($\psi=0$),
Theorem \ref{teorema} becomes an immediate consequence of the Brascamp--Lieb
inequality \cite{B-L}. 
On the other hand the extension
to bounded perturbations of a strictly convex 
function proved to be rather challenging. We refer the reader to
\cite{B-H,gentil-roberto,ledoux} and references therein to 
get an idea of the difficulties one has to face 
when leaving the purely convex setting.
Recently it was shown in \cite{LPY} that (\ref{vier}) holds
when $V$ is of the form $V(x)= a\,x^2 +\psi(x)$, $a>0$ and $\psi$ a 
bounded function. 
The authors prove the statement (\ref{vier}) by adapting the 
martingale method originally introduced in \cite{LY}. They also
prove that the stronger logarithmic Sobolev inequality holds. 
The recent paper \cite{Djalil} gives further development along
the same lines by slightly improving the hypothesis on the potential $V$. 
These results seem to rely strongly on the fact that $V$ 
is essentially quadratic.  Our approach 
is substantially simpler and covers a wider class of potentials.
On the other hand it is based on the permutation symmetry 
(exchangeability) of the canonical measure and it might be difficult 
to adapt to truly interacting non-product cases. 

\bigskip
The paper is organized as follows.
In the next section we use a uniform local central limit theorem
expansion to study a one--dimensional process which
plays a key role in the iterative proof of Theorem \ref{teorema}.
The latter is given in section 3. In section 4 we discuss the 
application to spectral gap estimates for Ginzburg-Landau 
processes. 

\subsection*{Acknowledgments}
I am greatly indebt to Fabio Martinelli. Many 
ideas and techniques involved in this note 
emerged from discussions with him during our joint work 
\cite{CapMa}. I also would like to thank Filippo Cesi, Cyril Roberto and 
Sebastiano Carpi for several interesting conversations.




\section{Uniform local CLT and an application}
We assume throughout that $V$ is a potential satisfying the hypothesis of
Theorem \ref{teorema}. Namely, $V(x)=\varphi(x)+\psi(x)$ and $\varphi''\geq\d >0$, $|\psi|_\infty < \infty$. Given $\r\in\bbR$ define the probability density
\begin{equation}
h^\r(x) = \frac{\nep{-V(x+\rho) -\l(\r) x}}{Z_\r}\,,
\label{hla}
\end{equation}
with $Z_\r=\int\nep{-V(x+\rho) -\l(\r) x}\dd x$. 
The parameter $\l=\l(\r)\in\bbR$, the so-called chemical potential,  
is uniquely determined by $\r$ through the condition
\be
\int x\,h^\r(x)\,\dd x = 0\,.
\la{la}
\end{equation}
We write $\si^2=\si^2_\r$ for the variance
$$
\si^2 = \int x^2\,h^\r(x)\,\dd x\,.
$$
Unless otherwise specified all integrals here and
below are understood to range over the real line. 
We call $\mu_\r$ the probability measure with density $h^\r(\cdot-\r)$. If 
$\mu_{N,\r}$ denotes the product 
$\mu_\r\otimes\cdots\otimes\mu_\r$ ($N$ times),
$N\in\bbN$, then the canonical measure $\nu_{N,\r}$ can be 
equivalently obtained as in (\ref{zwei}) with $\mu_N$ replaced by $\mu_{N,\r}$.

\subsection{Local central limit theorem}
Let $\pi_i$ be the canonical projection of $\bbR^N$ onto $\bbR$
given by $\pi_i\eta=\eta_i$, $\eta=(\eta_1,\dots,\eta_N)\in\bbR^N$. 
We call $\nu^1_{N,\r}$ the one-site marginal of $\nu_{N,\r}$, i.e.\
$\nu^1_{N,\r}=\nu_{N,\r}\circ\pi_1^{-1}$ is
the distribution of $\eta_1$ under $\nu_{N,\r}$.
By permutation symmetry all one-site marginals coincide.
The density
$g_{N,\r}$ of $\nu^1_{N,\r}$ 
can be written in the form
\be
g_{N,\r}(x) = 
\frac{h^\r(x-\r)G_{N-1}^\r(x-\r)}{G_N^\r(0)}\,,
\la{gnr}
\end{equation}
with 
\be
G_{N}^\r(x) = \int \dd\eta_1\cdots\int \dd\eta_N 
\Big(\prod_{i=1}^N h^\r(\eta_i)\Big)
\delta\Big(\sum_{i=1}^N\eta_i + x\Big)\,.
\la{GL}
\end{equation}
Here we are using Dirac's notation
$$
f(0) = \int \dd x f(x)\,\d(x)\,.
$$
Note that if we consider independent random variables $\eta_i$ with
common distribution defined by the density $h^\r$, then the normalized sum 
$$
% (\si^2N)^{-\frac12}
\frac1{\si\sqrt N}\sum_{i=1}^N \eta_i
$$
has a density given by 
$$
F_N^\r(z) = \si\sqrt{N}G_N^\r(-z\si\sqrt{N})\,.
$$ 
We shall use the classical local central 
limit theorem expansion for the density 
$F_N^\r$. 
Introduce the centered moments $m_k=m_k(\r)$: 
$$
m_k = \int x^k\,h^\r(x)\,\dd x\,, \quad \quad k=1,2,\dots
$$
so that in particular $m_1=0$, $m_2=\si^2$. 
A well known estimate (see \cite{Feller}, chap.\ XVI, Theorem 2) 
gives 
\be
F_N^\r(z)=
\frac{\nep{-\frac{z^2}{2}}}{\sqrt{2\pi}}
\Big(1+\frac{P_3(z)}{\sqrt{N}}+\frac{P_4(z)}{N}\Big) + 
O\big(N^{-\frac32}\big)
\la{expan2}
\end{equation}
where $P_3,P_4$ are the polynomials
$$
P_3(z) = \frac{m_3}{6\si^3}(z^3-3z)\,,\quad\quad 
P_4(z) = \frac{m_3^2}{72\si^6}(z^3-3z) + 
\frac{m_4 - 3\si^4}{24\si^4}(z^4 - 6z^2 + 3)\,.
$$
The above expansion holds as soon as the fifth moment $m_5$ exists.
What is important for us is that (\ref{expan2}) holds 
uniformly in $z\in\bbR$ and $\r\in\bbR$. This in turn follows 
directly from a uniform bound on the normalized 
moments $m_k/\si^k$, $k\leq 5$, which we prove below
as a consequence of a more general estimate. Note that since 
$|m_{k+1}|\leq \si\sqrt{m_{2k}}$ we can restrict to even powers. 

\begin{Le}
\la{kurt}
Assume 
$V=\varphi\,+\,\psi$ with 
$\varphi''\geq 0$, $|\psi|_\infty < \infty$. Then 
there exists $k=k(|\psi|_\infty)<\infty$ such that
for every $n\geq 2$
\be
\sup_{\r\in\bbR}
\Big|\frac{m_{2n}}{\si^{2n}}\Big|\,\leq \prod_{\ell=2}^n (1+k\ell^2)
%<\,\infty
\,\,.
\la{unifo}
\end{equation}
\end{Le}
\proof
We first establish a bound for the Poincar\'e constant in terms
of the variance $\si^2$. 
We denote by $\mu_\r$ the probability measure with density
$(\cZ_\r)^{-1}\nep{-V(x) - \l(\r)x}$ and by $\tilde \mu_\r$ 
the probability measure with density $(\tilde \cZ_\r)^{-1}
\nep{-\varphi(x) - \l(\r)x}$. Clearly $\cZ_\r = \nep{-\r\l(\r)} Z_\r$ 
(cf. (\ref{hla})) and
$$
\tilde \cZ_\r = \int  \nep{-\varphi(x) - \l(\r)x}\dd x
\in \big[\nep{-|\psi|_\infty}\cZ_\r, \nep{|\psi|_\infty}\cZ_\r\big]\,.
$$
Let $\g_\r$ and $\tilde\g_\r$ be the Poincar\'e constants associated
to $\mu_\r$ and $\tilde \mu_\r$ respectively. Let also $\tilde \si^2$
denote the variance of $\tilde \mu_\r$, 
$\tilde \si^2 = \tilde \mu_\r(x^2) - \tilde \mu_\r(x)^2$.
%Since $\varphi''\geq \d$
%$\tilde \mu_\r$ has a log-concave density and 
%it is easy to prove that $\tilde\g_\r\leq \d^{-1}$, see e.g.\ \cite{ledoux}. 
We shall use here a result derived by Bobkov in \cite{Bobkov},
which says that since the density of 
$\tilde \mu_\r$ 
is log-concave one has the bound 
\be
\tilde\g_\r \leq 12 \,\tilde \si^2
\la{bob}
\end{equation}
It is not difficult to establish a similar bound for $\mu_\r$. 
Namely, for any smooth function $f$ such that $\mu_\r(f)=0$ we write
\begin{align*}
\mu_\r(f^2) & \leq \mu_\r\big[(f-\tilde \mu_\r(f))^2\big] 
\leq \nep{2|\psi|_\infty} \tilde\mu_\r\big[(f-\tilde \mu_\r(f))^2\big] \\ 
&\leq \nep{2|\psi|_\infty} \,\tilde\g_\r \,\tilde\mu_\r\big[(f')^2\big]
\leq \nep{4|\psi|_\infty} \,\tilde\g_\r \, \mu_\r\big[(f')^2\big]\,,
\end{align*}
so that 
$$
\g_\r \leq \nep{4|\psi|_\infty} \tilde\g_\r\,.
$$
On the other hand
$$
\tilde \si^2 \leq \tilde \mu_\r\big[(x-\r)^2\big] \leq 
\nep{2|\psi|_\infty} \mu_\r\big[(x-\r)^2\big] = \nep{2|\psi|_\infty}\si^2\,.
$$
From (\ref{bob}) we obtain
\be
\g_\r \leq 12 \,\nep{6|\psi|_\infty} \,\si^2\,.
\la{bob1}
\end{equation}
Once we have such an estimate the proof of (\ref{unifo}) is immediate. 
For every $n\in\bbN$ we have
%$$
%m_{n+1} = \mu_\r\big[(x-\r)^{n+1}\big] \leq 
%\mu_\r\big[(x-\r)^{2}\big]^\frac12\,\mu_\r\big[(x-\r)^{2n}\big]^\frac12
%= \si_\r\,\sqrt{m_{2n}}\,. 
%$$
%In particular, we may restrict to even numbers $n$ to prove (\ref{unifo}).
%In this case we estimate
\begin{align*}
m_{2n} &= \var_{\mu_\r}\big[(x-\r)^n\big] + m_n^2\\
& \leq n^2 \,\g_\r\, \mu_\r \big[(x-\r)^{2(n-1)}\big] + m_n^2
= n^2 \,\g_\r\, m_{2(n-1)} + m_n^2\,.
\end{align*}
Setting $k=12\,\nep{6|\psi|_\infty}$, from (\ref{bob1}) we have
$$
m_{2n}\leq k\,n^2\si^2\, m_{2(n-1)} \,+ \, m_n^2\,.
$$
This implies the claim for $n=2$ and 
using $m_{n+1}^2\leq \si^2 m_{2n}$ one easily obtains the 
rest through induction. 
\qed


%%%%%%%%%%%%%%%%%%%%%%%%%%
%Since $m_3\leq \si\sqrt{m_4}$ and $m_5\leq \sqrt{m_4 m_6}$,
% (\ref{unifo}) follows from the following estimates.
%\b%e
%\sup_{\r\in\bbR}\,\,
%%\Big|
%\frac{m_{2n}}{\si^{2n}}
%%\Big|
%%\,<\,\infty
%\,\leq\,K_n
%\,.
%\la{kurt1}
%\end{equation}

\subsection{The operator $\cK$}
Here we introduce the relevant one--dimensional process
and prove a key spectral estimate, see Theorem \ref{key}.
Let $\cH$ denote the Hilbert space $L^2(\bbR,\nu_{N,\r}^1)$ and
use the symbol $\scalar{\cdot}{\cdot}$ for the corresponding scalar
product $\scalar{f}{g} = \nu_{N,\r}[(\bar f\circ\pi_1)( g\circ\pi_1)]$, 
with $\bar f$ denoting the complex conjugate function.
Write also $\av{f}$ for the mean of a function $f\in\cH$ 
w.r.t.\ $\nu_{N,\r}^1$. We write $\cH_0$ for the
subspace of $f\in\cH$ such that $\av{f}=0$. 
We define the stochastic self--adjoint 
operator $\cK:\cH\to\cH$ by
the sesquilinear form:
\be
\scalar{f}{\cK g} = \nu_{N,\r}\big[(\bar f\circ\pi_1)(g\circ\pi_2)\big]\,,\quad\quad
f,g\in\cH\,.
\la{opk}
\end{equation}
%If we denote by $\cF_i$, $i=1,\dots,N$ the $\si-$algebra generated by
%the projections $\pi_i$ we shall (with some abuse)
%identify the operator $\cK$ with a conditional expectation and write
%\be
%\cK f = \nu_{N,\r}\big(f\circ\pi_1\tc\cF_2\big)\,.
%\la{otherk}
%\end{equation}
Let $\xi_\r$ be the linear function 
$\xi_\r(x) = x-\r$. A simple computation shows that
\be
\cK\xi_\r = -\frac1{N-1}\,\xi_\r
\la{eig}
\end{equation}
for every $\r\in\bbR$. Thus the spectrum of $\cK$ always
contains the eigenvalues $-(N-1)^{-1}$ and $1$. We prove below that the
rest of the spectrum is confined around zero within a neighbourhood 
of radius $O(N^{-\frac32})$. 
\begin{Th}
\la{key}
There exists $C<\infty$ independent of $\r$ and $N$ 
such that for every $f\in\cH_0$ satisfying
$\scalar{f}{\xi_\r}=0$ one has
\be
\big|\scalar{f}{\cK f}\big|\leq  C\,N^{-\frac32}\,\scalar{f}{f}\,.
\la{key1}
\end{equation}
\end{Th}
The rest of this section deals with the proof of Theorem \ref{key}.
The idea is essentially the same as in \cite{CapMa} where 
this type of result has been established for a discrete lattice gas
model. 
%The details of the proof, however, are slightly different. 

\smallskip

Denote by $\tilde g_{N,\r}(x,y)$ the density of the joint distribution
of $(\eta_1,\eta_2)$ under $\nu_{N,\r}$. This is given by
\be
\tilde g_{N,\r}(x,y) = 
\frac{h^\r(x-\r)h^\r(y-\r)G_{N-2}^\r((x-\r)+(y-\r))}{G_N^\r(0)}\,.
\la{ggs}
\end{equation}
When $f$ is real and $\av{f}=0$ we write
\be
\scalar{f}{\cK f} = 
\int\int\dd x\,\dd y \,
g_{N,\r}(x) g_{N,\r}(y) Q_{N,\r}(x,y) f(x)f(y)
\la{Qs}
\end{equation}
where we introduced the kernel 
\be
Q_{N,\r}(x,y) = \frac{\tilde g_{N,\r}(x,y)- g_{N,\r}(x) g_{N,\r}(y)}
{g_{N,\r}(x) g_{N,\r}(y)}\,.
\la{Q}
\end{equation}
Define the set
\be
\cB_\r = \{(x,y)\in\bbR^2:\;\: |x-\r|+|y-\r|\leq \,B\,\si \log{N}\,\}
\la{bro}
\end{equation}
where $B$ is a constant to be fixed later on.
The following expansion is the key step in the proof of Theorem \ref{key}.
\begin{Le}
\la{exp}
For every $B<\infty$, there exists $C<\infty$ such that
\be
\sup_{\r\in\bbR}\;\sup_{(x,y)\in\cB_\r}\,
\left|\, Q_{N,\r}(x,y) + 
\frac{\xi_\r(x)\,\xi_\r(y)}{\si^2N}\, \right| \leq C\, N^{-\frac32}
\la{exp1}
\end{equation}
\end{Le}
\proof
In order to simplify notations we shall simply write 
$$
\bar x = \xi_\r(x) = x-\r \,,\quad\quad \bar y = \xi_\r(y) = y-\r\,.
$$ 
Using (\ref{gnr}) and (\ref{ggs}) we rewrite
\be
Q_{N,\r}(x,y)=
\frac{G_{N-2}^\r(\bar x+\bar y)G_N^\r(0) - 
G_{N-1}^\r(\bar x)G_{N-1}^\r(\bar y)}
{G_{N-1}^\r(\bar x)G_{N-1}^\r(\bar y)}
\,.
\la{exp01}
\end{equation}
We now use the expansion (\ref{expan2}). 
With the change of variable 
$$
\si\sqrt{N}G_N^\r(\bar x) = 
F_N^\r(-\bar x/\si\sqrt{N})
$$
we see that the only terms which are not negligeable w.r.t.\
$O(N^{-\frac32})$ in the range $|\bar x|\leq B\,\si\log N$ are 
given by the constant term in $P_4$ and the linear terms in $P_3$ and $P_4$. 
This implies 
\be
\sup_{\r\in\bbR}\;\sup_{|\bar x|\leq B\,\si\log{N}}
\left|\, 
\si\sqrt{N}G_N^\r(\bar x) - 
\frac{\nep{-\frac{\bar x^2}{2\si^2 N}}}{\sqrt{2\pi}}
 \Big(1+\frac{\al + \beta \bar x}{N} \Big) 
\, \right| 
\leq C\, N^{-\frac32}
\la{expan1}
\end{equation}
$$
\al:= \frac{m_4-3\si^4}{8\si^4}\, , 
\quad\quad\beta:=\frac{m_3}{2\si^4} + \frac{m_3^2}{24\si^7\sqrt{N}}\,.
$$
Note that by Lemma \ref{kurt} $\a$ is uniformly bounded and
$\beta \bar x$ is bounded by $C\,\log N$ in the range 
$|\bar x|\leq B\,\si\log{N}$ for some uniform $C<\infty$. 

\smallskip

\noindent
We introduce the following convention. We call $\e(N)$ anything which
vanishes at least as $O(N^{-\frac32})$ uniformly in 
$(x,y)\in\cB_\r$. Thus the result
%$|\bar x|\leq B\,\si\log{N}$ and $\r\in\bbR$
(\ref{expan1}) will be used in the form
\be
\si\sqrt{N}G_N^\r(\bar x) = 
\frac{\nep{-\frac{\bar x^2}{2\si^2 N}}}{\sqrt{2\pi}}
 \Big(1+\frac{\al + \beta \bar x}{N} \Big) \,+\,\e(N)\,.  
\la{accla}
\end{equation}
%We are ready to finish the proof.
Use now (\ref{accla}) to write
\begin{align*}
2\pi\si^2(N-1)&G^\r_{N-1}(\bar x)G^\r_{N-1}(\bar y) \\
&= 
\nep{-\frac{\bar x^2+\bar y^2}{2\si^2(N-1)}}
\Big(1+\frac{\al + \beta \bar x}{N-1} \Big)
\Big(1+\frac{\al + \beta \bar y}{N-1} \Big) \,+\,\e(N)  \\
& = 
\nep{-\frac{\bar x^2+\bar y^2}{2\si^2N}}
\Big(1+\frac{2\al + \beta(\bar x+\bar y)}{N} \Big) \,+\,\e(N)\,.
\end{align*}
Furthermore, writing $q(N)=(N-1)/\sqrt{N(N-2)}$
and observing that $q(N)=1+O(N^{-2})$, one has  
\begin{align*}
2\pi\si^2(N-1)&G^\r_{N-2}(\bar x+\bar y)G^\r_N(0) \\
&= 
q(N)\,
\nep{-\frac{(\bar x+\bar y)^2}{2\si^2(N-2)}}
\Big(1+\frac{\al + \beta (\bar x+\bar y)}{N-2} \Big)
\Big(1+\frac{\al}{N} \Big) \,+\,\e(N)  \\
& = 
\nep{-\frac{(\bar x+\bar y)^2}{2\si^2N}}
\Big(1+\frac{2\al + \beta (\bar x+\bar y)}{N} \Big)\,+\,\e(N) \\
& = 
\nep{-\frac{\bar x^2+\bar y^2}{2\si^2N}}\Big(1-\frac{\bar x\bar y}{\si^2N}\Big)
\Big(1+\frac{2\al + \beta (\bar x+\bar y)}{N} \Big)\,+\,\e(N)\,. 
\end{align*}
Inserting in (\ref{exp01}) we have obtained 
$$
Q_{N,\r}(x,y)
= -\frac{\bar x\bar y}{\si^2N}
\,+\,\e(N)\,.
$$
\qed

\smallskip
Next we need to control the atypical region $\cB_\r^c$. 
We shall use $\un_{\cB_\r}$ and $\un_{\cB_\r^c}$ to denote the indicator function
of $\cB_\r$ and its complement respectively. 
\begin{Le}
\la{tails}
There exist constants $C,B<\infty$
such that uniformly in $\r$
\be
\int\int \dd x\,\dd y\,
g_{N,\rho}(x)g_{N,\rho}(y)
\,\big|Q_{N,\r}(x,y)\big|^2\,\un_{\cB_\r^c}(x,y)
\leq 
C \,N^{-3}\,.
\la{tails1}
\end{equation}
\end{Le}
\proof
A first estimate gives 
\be
g_{N,\rho}(x)g_{N,\rho}(y)
\big|Q(x,y)\big|^2 \leq 2 g_{N,\rho}(x)g_{N,\rho}(y)\,+\, 
2\frac{\tilde g_{N,\rho}(x,y)^2}{g_{N,\rho}(x)g_{N,\rho}(y)}\,.
\la{tails20}
\end{equation}
%We need to prove that $\nu^1_{N,\r}$ has exponential tails 
%uniformly in $N,\r$. 
Observe that by (\ref{expan2}) -- see also (\ref{expan1}) -- we have
\be
\frac{G_{N-1}^\r(x-\r)}{G_N^\r(0)}
= \nep{-\frac{(x-\r)^2}{2\si^2(N-1)}}\,+\, O\big(N^{-\frac12}\big)
\la{expo}
\end{equation}
uniformly in $x\in\bbR$ and $\r\in\bbR$. 
In particular by (\ref{gnr}) we have 
$g_{N,\r}(x)\leq C\, h^\r(x-\r)$ uniformly. Then
$$
\int\int \dd x\,\dd y\,
g_{N,\rho}(x)g_{N,\rho}(y)
\,\un_{\cB_\r^c}(x,y) = \big[\nu^1_{N,\r}\otimes \nu^1_{N,\r}\big]
\big(\un_{\cB_\r^c}\big) \leq C^2\, 
\big[\mu_\r\otimes\mu_\r\big]\big(\un_{\cB_\r^c}\big)
$$
Now the measure $\mu_\r$ with density $h^\r(\cdot -\r)$ has exponential 
tails
and satisfies
\be
\big[\mu_\r\otimes\mu_\r\big]\big(\un_{\cB_\r^c}\big)
\leq 
C \,N^{-3}\,,
\la{expos}
\end{equation}
if $B$ is large enough, uniformly in $\r$. 
To prove (\ref{expos}) observe that if $\g_\r$ denotes the
Poincar\'e constant of $\mu_\r$ (see the proof of Lemma \ref{kurt}) 
then 
$$
\mu_\r\Big[\exp{\frac{|x-\r|}{\sqrt{\g_\r}}}\Big]<\infty
$$ 
uniformly in $\r\in\bbR$. The latter is obtained by iteration, 
applying Poincar\'e inequality to the functions $\nep{\pm (x-\r)/2}$.   
On the other hand by (\ref{bob1}) 
one has $\g_\r\leq k \si^2$ for some uniform 
constant $k<\infty$. This implies (\ref{expos})
and gives the correct estimate for 
the first term in (\ref{tails20}).

\smallskip

To handle the second term in (\ref{tails20})
we observe that 
$$
\frac{\tilde g_{N,\rho}(x,y)}{g_{N,\rho}(y)} = 
g_{N-1,\r_y}(x)\,,\quad\quad \r_y:=\frac{\r N - y}{N-1}\,.
$$
This can be checked using (\ref{gnr}), (\ref{ggs}) and the 
fact that for any $\r'$
$$
g_{N,\r}(x) = \frac{h^{\r'}(x-\rho')G_{N-1}^{\r'}
((x-\rho')+(\rho'-\rho)N)}
{G_{N}^{\rho'}((\rho'-\rho)N)}
$$
and similarly 
$$
\tilde 
g_{N,\rho}(x,y) = \frac{h^{\rho'}(x-\rho')
h^{\rho'}(y-\rho')G_{N-2}^{\rho'}
((x-\rho')+(y-\rho')+(\rho'-\rho)N)}
{G_{N}^{\rho'}((\rho'-\rho)N)}\,.
$$
The integral of
the second term in (\ref{tails20}) is then estimated 
as before 
%by
%$$
%\big[\nu^1_{N-1,\r_x}\otimes \nu^1_{N-1,\r_y}\big]
%\big(\un_{\cB_\r^c}\big)
%$$
%and 
observing that 
$|x-\r_y|+|y-\r_x|\geq (|x-\r|+|y-\r|)/2$ as soon as $N\geq 3$. 
Thus we have again the right tail estimate as above. \qed 


\bigskip

We are now able to finish the proof of Theorem \ref{key}.
Let us go back to (\ref{Qs}) and split the integral there as 
\begin{gather*}
\scalar{f}{\cK f} = \int\int\dd x\,\dd y \,
g_{N,\r}(x) g_{N,\r}(y) Q_{N,\r}(x,y) f(x)f(y)\un_{\cB_\r}(x,y)
\\
+ \int\int\dd x\,\dd y \,
g_{N,\r}(x) g_{N,\r}(y) Q_{N,\r}(x,y) f(x)f(y)\un_{\cB_\r^c}(x,y)\,.
\end{gather*}
The second term here
is estimated by Schwarz inequality and Lemma \ref{tails}
so that
\begin{align}
\Big|
\int\int&\dd x\,\dd y \,
g_{N,\r}(x) g_{N,\r}(y) Q_{N,\r}(x,y) f(x)f(y)\un_{\cB_\r^c}(x,y)
\Big|
\nonumber\\
&\leq \scalar{f}{f} 
\,\Big(\int\int \dd x\,\dd y\,
g_{N,\rho}(x)g_{N,\rho}(y)
\,\big|Q_{N,\r}(x,y)\big|^2\,\un_{\cB_\r^c}(x,y)\Big)^{\frac12}
\nonumber \\
& \leq 
C\,N^{-\frac32}\, \scalar{f}{f}\,.
\la{cm10}
\end{align}
The other term can be written as
\begin{gather*}
-\,\int\int\dd x\,\dd y \,
g_{N,\r}(x) g_{N,\r}(y) \,\frac{\xi_\r(x)\xi_\r(y)}{\si^2 N}\,
f(x)f(y)\un_{\cB_\r}(x,y) \\
+\,
\int\int\dd x\,\dd y \,
g_{N,\r}(x) g_{N,\r}(y) \Big[ 
Q_{N,\r}(x,y) + \frac{\xi_\r(x)\xi_\r(y)}{\si^2 N}
\Big] \,f(x)f(y)\un_{\cB_\r}(x,y)
\end{gather*}
A Scwharz inequality and Lemma \ref{exp} imply that 
the second term above is bounded by $C\,N^{-\frac32}\, \scalar{f}{f}$.
Since by assumption $\scalar{f}{\xi_\r}=0$ we rewrite the first term above
as
$$
\int\int\dd x\,\dd y \,
g_{N,\r}(x) g_{N,\r}(y) \,\frac{\xi_\r(x)\xi_\r(y)}{\si^2 N}\,
f(x)f(y)\un_{\cB_\r^c}(x,y)\,.
$$
We estimate the absolute value of this expression by 
$$
\scalar{f}{f}\,\frac1{\si^2N}\,\int_{|x-\r|\geq \frac{B}2\,\si\log{N}}
\dd x\, g_{N,\r}(x) \xi_\r(x)^2\,.
$$
Finally the last integral can be estimated using 
$g_{N,\r}(x)\leq C\,h^\r(x-\r)$ as in Lemma \ref{tails}
and the bound of Lemma \ref{kurt}:
$$
\int_{|x-\r|\geq \frac{B}2\,\si\log{N}}
\dd x \,g_{N,\r}(x) \xi_\r(x)^2 \leq C\,\sqrt{m_4} \;\Big( 
\int_{|x-\r|\geq \frac{B}2\,\si\log{N}}
\dd x \,h^\r(x-\r)\Big)^{\frac12}\leq C\,\si^2 N^{-\frac12}
$$
Collecting these estimates we have obtained
\be
\Big|
\int\int\dd x\,\dd y \,
g_{N,\r}(x) g_{N,\r}(y) Q_{N,\r}(x,y) f(x)f(y)\un_{\cB_\r}(x,y)
\Big| \leq 
C \,N^{-\frac32}\,\scalar{f}{f} \,.
\la{cm11}
\end{equation}
Estimates (\ref{cm10}) and (\ref{cm11}) imply the Theorem.





\section{Proof of Theorem \ref{teorema}}
The proof of Theorem \ref{teorema} is based on the recursive
inequality presented in the theorem below.
%Set  
%\be
%\o(N,\r)=\suptwo{f\in\cH:}{\av{f}=0}
%\,\frac{\scalar{f}{f}}{\scalar{f}{(\un-\cK)f}}
%\la{onr}
%\end{equation}
%where $\cK$ is defined by (\ref{opk}). 
Recall the definition (\ref{gamma}) of the Poincar\'e constant
$\g(N,\r)$, and set
\be
\g(N)=\sup_{\r\in\bbR} \g(N,\r)\,.
\la{gn}
\end{equation} 
\begin{Th}
\la{ccl}
There exist constants $C<\infty$ and $N_0\in\bbN$ 
such that for every $N > N_0$
\be
\g(N)\leq \big[1+CN^{-\frac32}\big]\,\g(N-1)\,.
\la{ccl1}
\end{equation}
\end{Th}
\proof
Take an arbitrary real smooth function
$F$ on $\bbR^N$. For simplicity we drop all subscripts and simply write
$\var(F)$ for $\var_{\nu_{N,\r}}(F)$ and $\cE(F)$ for $\cE_{N,\r}(F)$.
Let $\cF_k$ denote the $\si-$algebra generated by the 
one--site variables $\eta_k$, $k=1,\dots,,N$. 
$\var(F\tc\cF_k)$ denotes the $\cF_k$-measurable random variable
$\nu_{N,\r}(F^2\tc \cF_k)-\nu_{N,\r}(F\tc\cF_k)^2$.
We use the notation
$$
\cE^{(k)}(F) = \sum_{i:\,i\neq k} \nu_{N,\r}\big((\partial_iF)^2\big)\,.
$$
Note that 
$$
\sum_{k=1}^N\cE^{(k)}(F) = (N-1)\cE(F)\,.
$$
For any $F$ one has the decomposition
\be
\var(F) = \frac1N\sum_{k=1}^N
\nu\big(\var(F\tc\cF_k)\big) + 
\frac1N\sum_{k=1}^N \var\big(\nu(F\tc\cF_k)\big)\,.
\la{ccl2}
\end{equation}
By definition (\ref{gn}), for each $k$ we have
$$
\nu\big(\var(F\tc\cF_k)\big) \leq \gamma(N-1) \cE^{(k)}(F)\,.
$$
Summing over $k$ gives 
\be
\frac1N\sum_{k=1}^N
\nu\big(\var(F\tc\cF_k)\big) \leq \frac{N-1}N\,\g(N-1)\,\cE(F)\,.
\la{ccl3}
\end{equation}
We turn to estimate the second term in (\ref{ccl2}). 
Here comes the idea of \cite{CCL}. Namely assume without loss
that $\nu(F)=0$ and write the quadratic form
$$
\frac1N\sum_{k=1}^N \var\big(\nu(F\tc\cF_k)\big) = \nu\big(F\cP F)
$$
where the stochastic operator $\cP: L^2(\nu)\to L^2(\nu)$ 
is defined by 
$$
\cP F = \frac1N\sum_{k=1}^N \nu(F\tc\cF_k)\,.
$$
In this way (\ref{ccl2}) and (\ref{ccl3}) give
\be
\nu\big(F(\un-\cP)F\big) \leq \frac{N-1}N\,\g(N-1)\,\cE(F)\,.
\la{ccl35}
\end{equation}
We need a spectral
gap estimate for the generator $\un - \cP$. We are going to prove
\be
\nu\big(F(\un-\cP)F\big) \geq \frac{N-1}{N}\big[1-CN^{-\frac32}\big]
\,\nu\big(F^2\big)\,,
%\quad\quad F\in L^2(\nu)\,.
\la{gapp}
\end{equation}
for all real $F\in L^2(\nu)$ such that $\nu(F)=0$ with a uniform
constant $C<\infty$ independent of the density $\r$. 
Together with (\ref{ccl35}) this will complete the proof of the theorem.

\bigskip

Recalling the notation introduced in the previous section we
define the closed subspace $\Gamma$ of $L^2(\nu)$ consisting of sums of 
functions of a single variable:
\be
\Gamma = \Big\{F\in L^2(\nu): \; F = \sum_{k=1}^N f_k\circ\pi_k\,,\;
\: f_1,\dots,f_N\in\cH_0\,,\;\Big\}
\la{gaga}
\end{equation}
Since $\cP F\in\Gamma$ for every $F\in L^2(\nu)$ we may restrict to
$F\in \Gamma$ to prove (\ref{gapp}). For $F\in\Gamma$, 
$F = \sum_k f_k\circ\pi_k$, we define $\Phi_F = \sum_k f_k$, 
a function in $\cH_0$. 
Taking any real $F\in\Gamma$, a simple computation shows that
\be
\nu\big(F^2\big) = \scalar{\Phi_F}{\cK\Phi_F} + 
\sum_k \scalar{f_k}{(\un-\cK)f_k}\,,
\la{gag2}
\end{equation}
where $\cK$ is the operator defined in (\ref{opk}). 
Similarly one computes
\begin{align}
\nu\big(F(\un-\cP)F&\big)
=  \frac{N-2}N\,\scalar{\Phi_F}{\cK(\un-\cK)\Phi_F} \la{gag1} \\
& + \frac1N \sum_k \scalar{f_k}{(\un-\cK)[(N-1)\un + \cK]f_k}\,.
\nonumber
\end{align}
Consider now the subspace $\cS\subset\Gamma$ of symmetric functions:
\be
\cS =  \Big\{F\in L^2(\nu): \; F = \sum_{k=1}^N f\circ\pi_k\,,\;
\: f\in\cH_0\,\Big\}\,.
\la{ess}
\end{equation}
Since $\cS$ is invariant for $\cP$, i.e.\ $\cP\cS\subset\cS$ we may
consider separately the cases $F\in\cS$ and $F\in\cS^{\perp}$, with
$\cS^{\perp}$ denoting the orthogonal complement in $\Gamma$. 
When $F\in\cS$ we have $\Phi_F = N f$ and rearranging terms in (\ref{gag2})
and (\ref{gag1}) we obtain 
\begin{gather}
\nu\big(F^2\big) = N(N-1)\,\scalar{f}{[\cK+\frac\un{N-1}]f} \la{gag3} \\
\nu\big(F(\un - \cP)F\big) = (N-1)^2
\scalar{f}{[\un-\cK][\cK+\frac\un{N-1}]f} \la{gag4}
\end{gather}
By Theorem \ref{key} we see that $\cK+\frac\un{N-1}$ is non-negative 
on the whole subspace $\cH_0$. Moreover by (\ref{eig}) and (\ref{gag3}) we see
that $\nu(F^2) = 0$ when $f$ is a multiple of
$\xi_\r$. We may then restrict to the case $\scalar{f}{\xi_\r} = 0$. 
Writing $\tilde f = [\cK+\frac\un{N-1}]^\frac12 f$ and observing that
$\av{\tilde f} = 0$ and $\scalar{\tilde f}{\xi_\r} = 0$, Theorem \ref{key} 
yields the estimate
\begin{align}
\nu\big(F(\un - \cP)F\big) &\geq (N-1)^2\big[1-CN^{-\frac32}\big]\,
\scalar{\tilde f}{\tilde f} \nonumber\\
& = \frac{N-1}{N}\,\big[1-CN^{-\frac32}\big]
\,\nu\big(F^2\big)\,,\quad \quad F\in\cS\,.
\la{gappforS}
\end{align}

\smallskip

We turn to study the case $F\in\cS^\perp$. 
Let us first observe that in the definition (\ref{gaga}) of $\Gamma$
one can assume without loss that $\sum_k\scalar{f_k}{\xi_\r}=0$.
Indeed if $c = (N\scalar{\xi_\r}{\xi_\r})^{-1}\sum_k\scalar{f_k}{\xi_\r}$
and $g_k = f_k - c \xi_\r$, we have $\sum_k g_k\circ\pi_k = 
\sum_k f_k\circ\pi_k$ in $L^2(\nu)$ since by the conservation law 
$\sum_k \xi_\r\circ\pi_k = 0$. Therefore $\scalar{\Phi_F}{\xi_\r}=0$
may be assumed from the start.
Now, for every $G\in\cS$, 
$G=\sum_k g\circ\pi_k$, with $g\in\cH_0$ one has
$$
\nu(F G) = (N-1) \,\scalar{\Phi_F}{[\cK+\frac\un{N-1}]g}\,.
$$ 
Thus $F\in\cS^\perp$ implies that $[\cK+\frac\un{N-1}]\Phi_F$ is 
a constant in $\cH$. Since $\av{\Phi_F}=0$ and $\scalar{\Phi_F}{\xi_\r}=0$, 
Theorem \ref{key} implies $\Phi_F=0$. Writing
$\hat f_k = (\un-\cK)^\frac12 f_k$, then (\ref{gag2}) and (\ref{gag1})
imply 
\begin{gather}
\nu\big(F^2\big) = \sum_k \scalar{\hat f_k}{\hat f_k}\la{gag5} \\
\nu\big(F(\un - \cP)F\big) = \frac1N
\sum_k \scalar{\hat f_k}{[(N-1)\un + \cK]\hat f_k}
\la{gag6}
\end{gather}
Since $\av{\hat f_k}=0$ for all $k$ we may use Theorem \ref{key}
to estimate 
$$
\scalar{\hat f_k}{\cK\hat f_k}\geq -\frac1{N-1}
\scalar{\hat f_k}{\hat f_k}\,.
$$ 
From (\ref{gag5}) and (\ref{gag6}) we obtain
$$
\nu\big(F(\un - \cP)F\big)\geq \frac{N-2}{N-1} \nu\big(F^2\big)
= \frac{N-1}{N}\big[1 - \frac1{(N-1)^2}\big] \nu\big(F^2\big)\,.
$$
This ends the proof of the claim (\ref{gapp}). \qed


\bigskip

Once we have Theorem \ref{ccl} 
the conclusion of Theorem \ref{teorema} is straightforward.
Indeed, 
\be
%\sup_{\r\in\bbR}
\,\prod_{N=N_0+1}^\infty 
\big[1+CN^{-\frac32})\big]
\leq C'
\la{nzeroo}
\end{equation}
for some uniform constant $C'$ and 
Theorem \ref{ccl} yields
$$
\g(N)\leq C'\,\g(N_0)\,,\quad\quad N\geq N_0+1\,.
$$
The uniform Poincar\'e inequality of Theorem \ref{teorema}
then follows from the fact that $\g(N_0)$ is indeed finite.
\begin{Le}
For every $N_0\geq 2$ 
\la{nzero}
\be
\sup_{\r\in\bbR}\,\g(N_0,\r) <\infty\,.
\la{nzero1}
\end{equation}
\end{Le}
\proof
Let $\tilde \nu_{N,\r}$ denote the canonical measure 
obtained in (\ref{zwei}) where the potential $V$ is replaced by
its convex component $\varphi$. Let also $\tilde\g(N,\r)$ denote 
the corresponding Poincar\'e constant. Since $\varphi''\geq \d >0$ 
one can use the Brascamp--Lieb inequality \cite{B-L} to prove 
$\tilde\g(N,\r)\leq \d^{-1}$, uniformly in $N$ and $\r$, see also 
\cite{Cap}.
A standard argument (as in the proof of Lemma \ref{kurt}) on the other hand
gives $\g(N,\r)\leq \nep{4N|\psi|_\infty} \tilde\g(N,\r)$, for every 
$N\in\bbN$ and $\r\in\bbR$. This gives, uniformly in $\r$ 
$$
\g(N,\r)\leq \d^{-1} \nep{4N|\psi|_\infty}\,.
$$
\qed






\section{Ginzburg-Landau processes}

We consider the discrete lattice
$\bbZ^d$, with $d\geq 1$ an integer. 
Given a finite subset $\La\subset\bbZ^d$, we denote by $\La^*$ the set
of oriented bonds $b$ contained in $\La$, i.e.\ the couples 
$b=(x,y)$, $x,y\in\La$ with $x=y+e$, $e$ a unit vector in $\bbZ^d$. 
Denoting $\La_L= \{1,2,\dots,L\}^d$, the $L-$hypercube in $\bbZ^d$,
we define the product measure $\mu_{\La_L,\r}$ as the 
usual grand canonical measure $\mu_{N,\r}$ with $N=L^d$. 
Then $\nu_{\La_L,\r}$ stands for the probability measure 
obtained from $\mu_{\La_L,\r}$ by conditioning on 
$L^{-d}\sum_{x\in\La_L}\eta_x=\r$.  
The Ginzburg-Landau dynamics is defined by the Dirichlet form
\be
\cD_{L,\r}(F)= \frac12\sum_{b\in\La_L^*}\nu_{\La_L,\r}
\big[(\grad_bF)^2\big]
\la{sette}
\end{equation}
where we used the notation
$$\grad_bF=\partial_yF-\partial_xF\,,\;\;\;b=(x,y)\,.$$
%Below we restrict to the class of smooth functions which only depend on
%bond differences of the configuration $\eta\in\bbR^{\La_L}$. 
%More precisely, we define
%\be
%\cA_L = \Big\{f\in\cC^\infty(\bbR^{\La_L})\,:
%\;\;\partial_yf=0
%\;\;\;{\rm for\;some\;}y\in\La_L\Big\}\,.
%\la{otto}
%\end{equation}
The inverse of the spectral gap associated to $\cD_{L,\r}$ is given by
\be
\chi(L,\r)=\sup_{F}\frac{\var_{\nu_{\L_L,\r}}(F)}
{\cD_{L,\r}(F)}
\la{gammat}
\end{equation}
with the supremum ranging over all real smooth functions on $\bbR^{\L_L}$.
As already observed in \cite{Cap} 
we have a simple upper bound on $\chi(L,\r)$ in terms 
of $\g(N,\r)$ with $N=L^d$.

\begin{Le}
\la{paths}
There exists a constant $C$ only depending on $d$ such that  
\be
\chi(L,\r) \leq C\,L^2\,\g(L^d,\r)
\la{paths1}
\end{equation}
\end{Le}
\proof
We first make some observations about paths in $\La_L$. 
We denote $\cC_{xy}(L)$ the set of all paths $\gamma_{xy}$ connecting 
sites $x,y\in\La_L$, which use only bonds in $\La_L^*$. The length
of a path, denoted $|\gamma_{xy}|$ is the number of bonds composing it.
Given $x,y\in\La_L$
we need a rule to select a single path $\gamma_{xy}$ from $\cC_{xy}(L)$. 
We may choose $\gamma_{xy}$ as follows. Fix $x,y\in\La_L$ and 
define points $x^{(i)}$, $i=0,\dots,d$, such that $x^{(0)}=x$, $x^{(d)}=y$,
and when $i=1,\dots,d-1$ 
$$x^{(i)}_j=
\begin{cases}y_j & j=1,\dots,i\\
x_j & j=i+1,\dots,d
\end{cases}$$ 
Call $\gamma^{(i)}$, $i=1,\dots,d$, 
the straight line parallel to the $i$-th axis joining sites $x^{(i-1)}$
and $x^{(i)}$. The path $\gamma_{xy}$ is given by 
$\gamma^{(1)}\cup\cdots\cup\gamma^{(d)}$. 
It is not difficult to prove the following properties: 
there exists a finite constant $k$ only depending on $d$
such that
\begin{itemize}
\item{for every $x,y\in\La_L$, $|\gamma_{xy}|\leq kL$\,, and}
\item{for every $b\in\La_L^*$, $\sum_{x,y\in\La_L}1_{\{b\in\gamma_{xy}\}}
\leq kL^{d+1}$}
%\item{for every $x\in\La_L$, $\sum_{y,z\in\La_L}
%\sum_{b\in\La_L^*}1_{\{b\in\gamma_{zy}\}}
%1_{\{b\in\gamma_{xy}\}}\leq kL^{d+2}$}
\end{itemize}
%%
%
%there are at most
%$kL$ sites $x\in\La_L$ with $\gamma_{0x}\ni b$.} 
%\end{itemize}
%To prove that the above choice is possible one can 
When we write $\gamma_{xy}$ below we always assume that this path has been 
chosen according to the above rule. 

\smallskip

Given $\eta\in\bbR^{\La_L}$, $y\in\La_L$ we write $\eta^{(y)}$ for
the configuration
$$\eta^{(y)}_x=\begin{cases}
\eta_x & x\neq y\\
\r L^d-\sum_{z\neq y}\eta_z & x=y \end{cases}\,.$$
For $F:\bbR^{\La_L} \to \bbR$, we denote $F_y(\eta)$ the function
$\eta\to F(\eta^{(y)})$. Clearly, for any $y\in\La_L$ we have
\be
\var_{\nu_{\La_L,\r}}(F)=\var_{\nu_{\La_L,\r}}(F_y)\,.
\la{fy}
\end{equation}
It is then
sufficient to show
\be
L^{-d}\sum_{y\in\La_L}\sum_{x\in\La_L}\nu_{\La_L,\r}
\big[(\partial_xF_y)^2\big]\leq C\,L^2 \,\cD_{L,\r}(F)\,.
\la{diciotto}
\end{equation}
For any $x,y\in\La_L$ we have
$$\partial_xF_y = \big(\partial_xF\big)_y - \big(\partial_yF\big)_y$$
and therefore
$$\nu_{\La_L,\r}
\big[(\partial_xF_y)^2\big] = \nu_{\La_L,\r}
\big[(\partial_xF - \partial_yF)^2\big]\,.$$
We write 
$$\partial_xF - \partial_yF = 
\sum_{b\in\gamma_{yx}}\grad_bF\,.$$
Since $|\gamma_{yx}|\leq kL$, 
Schwarz' inequality gives
\be
\big(\partial_xF - \partial_yF\big)^2
\leq kL\sum_{b\in\La_L^*} \big(\grad_bF\big)^2
1_{\{b\in\gamma_{yx}\}}\,.
\la{venti}
\end{equation}
From the second property of our paths 
we see that (\ref{diciotto}) with $C=k^2$ follows 
from (\ref{venti}) when summing over $x,y$ and dividing by $L^d$. 
\qed

\bigskip

From Theorem \ref{teorema} we obtain
\begin{Cor}
\la{corollario}
Assume $V=\varphi+\psi$ with $\varphi\in\cC^2$, $\varphi''\geq \d>0$ and 
$|\psi|_\infty <\infty$. Then there exists $C=C(\d,|\psi|_\infty)<\infty$ 
such that for every $\r\in\bbR$ and $L\in\bbN$
\be
\var_{\nu_{\L_L,\r}}(F)\leq C\,L^2\,
\cD_{L,\r}(F)
\la{coro}
\end{equation}
holds for every smooth function $F$ on $\bbR^{\L_L}$. 
\end{Cor}



















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Pitman Res. Notes Math. Ser., 283, Longman Sci.\ Tech.\ , 1993.

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%Nongradient models in hydrodynamic scaling.} 
%Analysis, geometry and probability, 397--416, Texts Read. Math., 10, 
%Hindustan Book Agency, Delhi, 1996






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