%\documentstyle[12pt]{article}
\documentstyle[11pt,amscd,amssymb]{amsart}
\setlength{\textheight}{9.1in}
\addtolength{\textwidth}{10em}
\voffset -0.8in
\hoffset -6em
\newcommand{\N}{{\Bbb N}}
\newcommand{\Ztwo}{{\Bbb Z}^2}
\newcommand{\Ztwol}{{\Bbb Z}^2_l}
\newcommand{\e}{\mbox{e}}
\newcommand{\E}{{\Bbb E}}
\renewcommand{\P}{{\Bbb P}}
\newcommand{\R}{{\Bbb R}}
\newcommand{\df}{\ \stackrel{\Delta}{=}\ }
\newcommand{\iso}{\approx}
\newcommand{\so}{{\small o}(1)}
\newcommand{\dr}{\text{d}}
\begingroup % Confine the \theorembodyfont command
%\theorembodyfont{\sl}
\newtheorem{bigthm}{Theorem} % Numbered separately, as A, B, etc.
\newtheorem{thm}{Theorem}
%[section] % Numbered within each section
\newtheorem{cor}[thm]{Corollary} % Numbered along with thm
\newtheorem{lem}[thm]{Lemma} % Numbered along with thm
\newtheorem{prop}[thm]{Proposition} % Numbered along with thm
\endgroup
\renewcommand{\thebigthm}{\Alph{bigthm}} % Number as "Theorem A."
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition} % Numbered along with thm
\theoremstyle{remark}
\newtheorem{rem}[thm]{Remark} % Numbered along with thm
\newtheorem{ex}[thm]{Example} % Numbered along with thm
\newtheorem{notation}{Notation}
%\renewcommand{\theequation}{\thesection.\arabic{equation}}
\renewcommand{\thenotation}{} % to make the notation
% environment unnumbered
\newtheorem{terminology}{Terminology}
\renewcommand{\theterminology}{} % to make the terminology
% environment unnumbered
\begin{document}
\title[\,]{A note on the decay of correlations
Under $\delta-$Pinning}
%\maketitle
%\begin{center}
\author{Dmitry Ioffe}
\address{
WIAS, Mohrenstr.~39
D-10117 Berlin, Germany and Faculty of Industrial Engineering,
Technion, Haifa 32000, Israel}
%\end{center}
%}
%\newline
%and\newline
%Department of Industrial Engineering,\newline
%Technion, Haifa 3200, Israel}
\author{ Yvan Velenik}
\address{Fachbereich Mathematik, Sekt.~MA~7-4, TU-Berlin,
Strasse~des~17 Juni 136, D-10623 Berlin, Germany}
\date{\today}
\begin{abstract}
We prove that for a class of massless $\nabla\phi$ interface models on $\Ztwo$
an introduction of an arbitrary small pinning self-potential leads to
exponential decay of correlation, or, in other words, to creation of mass.
\end{abstract}
\maketitle
%\footnote{}
%%% In the address, show linebreaks with double backslashes:
%\address{D\'{e}partment de Math\'{e}matiques,\newline
%EPF-L, CH-1015 Lausanne,\newline
%Switzerland}
%%% Email address is optional. If you include it, use a double at
%%% sign "@@" to produce a single at sign in the printed copy, e.g.,
%%% \email{nsteenrod@@math.princeton.edu}
%\email{ioffe@pascal.wias-berlin.de}
%\email{\,}
%%% To have the current date inserted, use \date{\today}:
\vskip 0.1in
In this note we study a family of effective interface models over $\Ztwo$ with
the formal Hamiltonian ${\cal H}$ given by
\begin{equation}
\label{hamiltonian}
{\cal H}(\phi )~=~\sum_{i\sim j} V(\phi_i -\phi_j ),
\end{equation}
where the summation is over all nearest neighbours $i\sim j$ of $\Ztwo$, and
the following two assumptions are made on the interaction potential $V$:
\begin{itemize}
\item $V$ is even and smooth
\item There exists a constant $c_V \geq 1$, such that
\begin{equation}
\label{V}
\frac1{c_V}~\leq~V^{\prime\prime} (t)~\leq~c_V\qquad\qquad\forall
t\in\R .
\end{equation}
\end{itemize}
\begin{rem}
No further assumptions on $c_V$ are made, and, in fact, we expect that the
results of the paper remain true if only the lower bound in \eqref{V} is
assumed. Also, though we do not stipulate it explicitly at each particular
instance, the values of {\bf all} the positive constants we use below depend on
$c_V$.
\end{rem}
Given a set $A\subset\Ztwo$ with a finite complement $A^c\df\Ztwo\setminus A$,
we use $\P_A$ to denote the finite volume Gibbs measure on $\Omega_A\df
\R^{A^c}$ with the Hamiltonian ${\cal H}$ and zero boundary conditions on
$A$;
\begin{equation}
\label{PA}
\P_A (\dr \phi )~=~\frac1{{\bf Z}(A)}\e^{-{\cal H}(\phi )}\prod_{i\in
A^c}
\dr h_i\prod_{j\in A}\delta_0 (\dr h_j ) .
\end{equation}
It is well known that $\P_A$ delocalizes as $A^c\nearrow\Ztwo$; maybe the
easiest way to see this is to use the reverse Brascamp-Lieb inequality
\cite{DGI} which implies that the variance of $\phi_0$ under $\P_A$ dominates
the
corresponding Gaussian variance. If, however,
an, essentially arbitrary small, pinning self-potential is added to ${\cal H}$,
then the situations radically changes, and the infinite volume Gibbs state
exists in the usual sense. This phenomenon has been first worked out in the
Gaussian case ($c_V =1$) in \cite{DMRR}. Our main reference \cite{DV} contains
a proof of the localization for a fairly general class of interactions and
self-potentials. In this note we prove that in the case of the family of random
interfaces as in \eqref{hamiltonian}, the delocalization/localization
transition is sharp in the sense that it always comes together with the
exponential decay of correlations, or, using the language of a more physically
oriented literature, with the creation of mass.
For simplicity, but also in order to give a cleaner exposition of otherwise
more general renormalization ideas behind the proof, we consider here only the
case of the so called $\delta$-pinning, thereby generalizing recent results of
\cite{BB} on purely Gaussian fields (that is again $c_V=1$):
Given a box $\Lambda_N\df [-N,...,N]^2\subset\Ztwo$ and a number $J\in\R$
(which characterizes the strength of the pinning) we define the following
measure $\hat{\P}_N$ on $\R^{\Lambda_N}$:
\begin{equation}
\label{PNhat}
\hat{\P}_N (\dr \phi )~=~\frac1{\hat{{\bf Z}}_N}\e^{-{\cal H}(\phi )}
\prod_{i\in\Lambda_N}\left(\dr \phi_i +\e^J\delta_0(\dr \phi_i
)\right)
\prod_{j\in\Ztwo\setminus\Lambda_N}\delta_0 (\dr \phi_j ) .
\end{equation}
Notice that the case $J= -\infty$ corresponds to the original measure on
$\R^{\Lambda_N}$ with the Hamiltonian \eqref{hamiltonian}, which delocalizes as
$N\to\infty$.
\begin{lem}
\label{Lemma1}
For every $J\in\R$ there exists an exponent (mass) $m=m(J) >0$ and a
constant $c_1 = c_1 (J)<\infty$, such that
\begin{equation}
\label{mass}
{\Bbb C}\text{\rm ov}_{\hat{\P}_N}\big( \phi_i
;\phi_j\big)~\leq~c_1\e^{-m\|i-j\|}
\end{equation}
uniformly in $N$ and in $i,j\in\Ztwo$.
\end{lem}
Of course, there is nothing to prove if either $i$ or $j$ lies outside of
$\Lambda_N$. In fact, the sub-index $N$ is superfluous - all the estimates we
use and obtain simply do not depend on a particular $\Lambda_N$, and the only
reason we need it is to make the definitions mathematically meaningful. From
now on we shall drop the sub-index $N$ from the notation.
\vskip 0.1in
A right way to think about \eqref{PNhat} is as of the joint distribution of the
field of random interface heights $\{\phi_i\}_{i\in\Ztwo}$ and the random
``dry'' set ${\cal A}$;
$$
{\cal A}~\df~\left\{ i\in\Ztwo :
\phi_i =0\right\} .
$$
Integrating out all the height variables $\phi$ in \eqref{PNhat} we arrive to
the following probability distribution for ${\cal A}$;
\begin{equation}
\label{dryset}
\hat{\P}\left( {\cal A}=A\right)~\df~\rho (A)~=~\frac1{\hat{\bf Z}}
\e^{J|A|}{\bf Z} (A)~=~\frac{\e^{J|A|}{\bf Z} (A)}
{\sum_{D}\e^{J|D|}{\bf Z} (D)} ,
\end{equation}
where the partition function ${\bf Z}(A)$ is the same as in \eqref{PA}.
Using the probabilistic weights $\{\rho (A)\}$ one can rewrite $\hat{\P}$ as
the convex combination,
\begin{equation}
\label{Phatconv}
\hat{\P}(\cdot )~=~\sum_A\rho (A)\P_A (\cdot ) .
\end{equation}
Since under each $\P_A$ the distribution of $\phi_i$ is symmetric for every
$i\in\Ztwo$, this gives rise to the following decomposition of the
covariances:
\begin{equation}
\label{Covconv}
{\Bbb C}\text{ov}_{\hat{\P}}\big( \phi_i ;\phi_j\big)~=~
\sum_{A}\rho (A)\langle \phi_i ;\phi_j\rangle_{A} .
\end{equation}
At this point we shall utilize the random walk representation of $\langle
\phi_i ;\phi_j\rangle_{A}$ which has been first developed in the PDE context in
\cite{HS}. We follow the approach of \cite{DGI}, where the
Helffer-Sj\"{o}strand representation was put on the probabilistic tracks:
One constructs a stochastic process $\big(\Phi (t) ,X(t)\big)$, where:
\begin{itemize}
\item $\Phi (\cdot ) $ is a diffusion on $\R^{A^c}$ with the invariant measure
$\P_A$.
\item Given a trajectory $\phi (\cdot )$ of the process $\Phi$, $X(t)$ is an,
in general inhomogeneous, transient random walk on $A^c\cup\partial
A^c\subset\Ztwo$ with the life-time
$$
\tau_A~\df~\inf\{t:~X(t)\in A\},
$$
and the time-dependent jump rates
\begin{equation}
\label{rates}
a(i,j;t)~=~\left\{
\begin{split}
&V^{\prime\prime}\big(\phi_i (t) -\phi_j (t)\big),\qquad \text{if}\
i\sim j\\
&0,\qquad\qquad\qquad\ \qquad \text{otherwise}
\end{split}
\right.
\end{equation}
\end{itemize}
Let us use ${\cal E}_{i,\phi}^{A}$ to denote the law of $\big( X(t) , \Phi
(t) \big)$ starting from the point $(i,\phi )\in A^c\times \R^{A^c}$. Then
(\cite{HS},\cite{DGI}),
\begin{equation}
\label{RWrepr}
\langle\phi_i ,\phi_j\rangle_A~=~\left\langle
{\cal E}_{i,\phi}^{A}\int_{0}^{\tau_A}{\Bbb I}_{\{X(s)=j\}}\dr
s\right\rangle_A.
\end{equation}
Substituting the latter expression into \eqref{Covconv},
\begin{equation}
\label{basic}
{\Bbb C}\text{ov}_{\hat{\P}}\big( \phi_i ;\phi_j\big)~=~
\sum_A\rho (A)
\left\langle
{\cal E}_{i,\phi}^{A}\int_{0}^{\tau_A}{\Bbb I}_{\{X(s)=j\}}\dr
s\right\rangle_A.
\end{equation}
It is very easy now to explain the logic behind the proof of
Lemma~\ref{Lemma1}: The expression
$$
{\cal E}_{i,\phi}^{A}\int_{0}^{\tau_A}{\Bbb I}_{\{X(s)=j\}}\dr s
$$
describes the time spent by the random walk $X(\cdot )$ starting at $i$ in the
site $j$ before being killed upon entering the dry set $A$ which, for the
purpose, could be considered as a random killing obstacle. In order to prove
that this time is exponentially (in $\| i-j\|$) small one needs an appropriate
density estimate on $A$ and a certain path-wise control on the exit
distributions of $X (\cdot )$. In the Gaussian case considered in \cite{BB},
$X(\cdot )$ happens to be just the simple random walk on $\Ztwo$ which is
completely decoupled from the diffusion part $\Phi (\cdot )$, and, thus,
behaving independently of $A$ and the initial condition $\phi\in\R^{A^c}$.
This lead in \cite{BB} to a resummation argument, which substantially
facilitated the matter. One of the main difficulties in the non-Gaussian case
we consider here is the dependence of the distribution of $X(\cdot )$ on the
realization of the dry set $A$ and on the sample path of the diffusion $\Phi$.
We still have very little to say about this dependence. However, due to the
basic assumption \eqref{V} on the interaction potential $V$, the jump rates
$a(i,j;t)$ in \eqref{rates} are uniformly bounded above and below:
\begin{equation}
\label{jumprates}
\frac1{c_V}~\leq a(i,j ;t)~\leq~c_V .
\end{equation}
In particular one always has a rough control over probabilities of hitting
distributions. For example, if the random walk $X$ enters a box ${\bf B}_l$ of
linear size $l$ which is known to contain a dry site; it would be convenient to
call such a box ``dirty'', then the probability that $X$ hits this site (and
consequently dies there) before leaving ${\bf B}_l$ should be bounded below by
some positive number $p=p(l)>0$. Thus if the realisation $A$ of the random dry
set ${\cal A}$ is such, that on its way from $i$ to $j$ the walk $X$ cannot
avoid visiting less than $\epsilon \| i-j\|$ disjoint dirty $l$-boxes, the
probability that it eventually reaches $j$ before being killed should be
bounded above by something like
$$
\left( 1- p(l) \right)^{\epsilon \| i-j\|} .
$$
Proposition~\ref{Prop2} below makes this computation precise.
The crux of the matter, however, is to ensure that on a certain finite
$l$-scale the density of the dirty $l$-boxes is so high, that only with
exponentially small probabilities the realization $A$ of ${\cal A}$ enables an
$\epsilon$-clean passage from $i$ to $j$. A statement of this sort is given
in Proposition~\ref{Prop1}.
\vskip 0.1in
\noindent
Once the renormalization approach sketched above is accepted as the strategy of
the proof, the first drive of an associative thinking is to try to compare the
distribution of ${\cal A}$ on different $l$-scales with, say, independent
Bernoulli percolation or other known models with controllable decay of
connectivities. This we have tried and failed, and, at least in the case of
$\Ztwo$, such a comparison is unlikely.
The relevant statistical properties of the random dry set ${\cal A}$ on
various finite length scales are captured in the following estimate which
generalizes the key Proposition~5.1 in \cite{DV}
\begin{thm}
\label{Main}
For each $J\in \R$ there exists a number $R =R (J)<\infty$ and
exponent $\nu =\nu (J) >0$, such that whenever a
finite set $B\subset\Ztwo$ admits a decomposition
\begin{equation}
\label{Bdecomp}
B~=~\bigvee_{l=1}^n B_l
\end{equation}
into connected disjoint components $B_1,...,B_n$ with
\begin{equation}
\label{diam}
\text{diam}\big( B_l\big)~\geq~R;\qquad\qquad l=1,...,n ,
\end{equation}
the following exponential upper bound on having all of $B$ ``clean of
dry points''
holds:
\begin{equation}
\label{mainbound}
\sum_{A\cap B =\emptyset}\rho (A)~\leq~\e^{-\nu |B|}.
\end{equation}
\end{thm}
We relegate the proof of Theorem~\ref{Main} to the end of the
paper, and, assuming for the moment its validity, directly proceed to
the proof of the mass-generation claim of Lemma~\ref{Lemma1}.
\vskip 0.2in
\noindent
{\it Proof of Lemma~\ref{Lemma1}:\ }\ The number $R=R(J)$ which appears in
the basic Theorem~\ref{Main} sets up the stage for the finite scale
renormalization analysis of the random dry set ${\cal A}$. Let us pick a
number $l>R;\ l\in\N ,$ and define the renormalized lattice
$$
\Ztwol~\df~(2l+1)\Ztwo .
$$
To distinguish between the sets on the original lattice $\Ztwo$ and those on
the renormalized one $\Ztwol$ we shall always mark the latter by the
super-index $l$. For example ${\bf B}^l (x,r)$ stands for the $\Ztwol$ lattice
box centered at $x\in\Ztwol$;
$$
{\bf B}^l (x,r)~\df~\left\{ y\in\Ztwol :\ \ \|x-y\|\leq lr\right\} .
$$
Let us define $\Gamma^l (r)$ as the set of all $\Ztwol$-nearest neighbour
lattice paths leading from the origin to the boundary $\partial {\bf B}^l
(x,r)$. With each $\gamma^l\in \Gamma^l (r)$ we associate a connected chain
$\tilde{\gamma}^l$ of $l$-blocks on the original lattice $\Ztwo$;
$$
\tilde{\gamma}^l~\df~\bigcup_{x\in\gamma^l} {\bf B} (x,l) .
$$
Let us fix a number $\epsilon \in (0,1)$. We say that a path
$\gamma^l\in\Gamma^l$ is $(r,\epsilon )$-clean in $A\subset\Ztwo$, if
$$
\#\left\{x\in\gamma^l :\ \ {\bf B} (x,l)\cap A \neq\emptyset\right\}~<~
\epsilon r .
$$
Similarly, we say that a set $A\subset\Ztwo$ is $(r,\epsilon )$-clean if there
exists a path $\gamma^l\in\Gamma^l (r)$ which is $(r,\epsilon )$-clean for
$A$. Otherwise, we shall call $A$ $(r,\epsilon)-$dirty.
\begin{prop}
\label{Prop1} For each $\epsilon\in (0,1)$ there exist a number $l_0 =l_0
(\epsilon, J)<\infty$ and a radius $r_0 =r_0 (\epsilon )$, such that for every
choice of $l\geq l_0$;
$$
\sum_{A~\text{is}~(r,\epsilon )-\text{clean}}\rho\left(
A\right)~\leq~
\e^{-c_2 (\epsilon ,l)r} ,
$$
uniformly in $r\geq r_0$, where $c_2 (\epsilon ,l )$ diverges (as $l^2$) with
$l$.
\end{prop}
\vskip 0.1in
\noindent
{\it Proof:}\ \
The condition on $r_0(\epsilon)$ is a semantic one - the only thing we want is
to ensure that $r >[\epsilon r]$.
Let us estimate the probability of the event $\{ A~\text{is}~(r,\epsilon
)-\text{clean}\}$ as follows:
\begin{equation}
\label{epsilonclean}
\sum_{A\,\text{is}\,(r,\epsilon )-\text{clean}}\rho\left(
A\right)~\leq~
\sum_{k=r}^{\infty}\sum_{\gamma^l\in\Gamma^l :|\gamma^l |=k}\ \
\sum_{A\,:\gamma_l\,\text{is}\,(r,\epsilon)-\text{clean in}\,A}\rho
(A) .
\end{equation}
Each path $\gamma^l =(0,x_1,...,x_k);\ \gamma^l\in\Gamma^l$, which is
$(r,\epsilon )$-clean in $A$ contains at most $[\epsilon r]$ vertices
$x_{i_1},...,x_{i_M};\ M\leq [\epsilon r]$, such that the corresponding
$l-$blocks have a non-empty intersection with $A$;
$$
{\bf B} (x_i,l)\cap A~\neq~\emptyset ;\qquad i=1,...,M .
$$
%There are at most
%\begin{equation}
%\label{dirtyfactor}
%\sum_{M=0}^{[\epsilon r]}\left(
%\begin{split}
%&\ k\\&M
%\end{split}
%\right)~\leq~
%[\epsilon r]
%\left(
%\begin{split}
%&\ k\\&[\epsilon r ]
%\end{split}
%\right)
%\end{equation}
Whatever happens, for a path $\gamma^l$ of length $k$ there are at most $2^k$
(in fact much less due to the restriction $M\leq [\epsilon r]$) possible ways
to choose a sub-family $\tilde{\gamma}_{\text{dirty}}^l$;
$$
\tilde{\gamma}_{\text{dirty}}^l~\df~\bigcup_{i=1}^{M}{\bf B}(x_i ,l),
$$
of ``dirty'' block along $\tilde{\gamma}^l$. On the other hand, fixing both
$\tilde{\gamma}^l$ and its ``dirty part'' $\tilde{\gamma}^l_{\text{dirty}}$,
we can use Theorem~\ref{Main} to obtain
\begin{equation}
\label{cleanfactor}
\sum_{A\cap\tilde{\gamma}^l\setminus\tilde{\gamma}^l_{\text{dirty}}=
\emptyset}\rho (A)~\leq~
\exp\{-\nu |
\tilde{\gamma}^l\setminus\tilde{\gamma}^l_{\text{dirty}}|\}~\leq~
\e^{-\nu (k-[\epsilon r])l^2}
\end{equation}
We, thus, conclude,
that for any $k\geq r$ and for each
$\gamma^l\in\Gamma^l$ with $|\gamma^l | =k$,
\begin{equation*}
\sum_{A:\,\gamma_l~\text{is}\,(r,\epsilon)-\text{clean in}\,A}\rho
(A)~\leq~
%\left(
%\begin{split}
%&\ k\\&[\epsilon r ]
%\end{split}
%\right)
\e^{-\nu (J) l^2 (k-[\epsilon r]) +k\log 2} .
\end{equation*}
Using the above estimate together with the trivial bound;
$$
\#\left\{ \gamma^l\in\Gamma^l :\ |\gamma^l |=k\right\}~\leq~4^k,
$$
to perform the summation in \eqref{epsilonclean} we arrive at the claim of
Proposition~\ref{Prop1}.
\qed
\vskip 0.1in
\noindent
Nothing in the above argument depends on the fact that the box ${\bf B}(0,rl)$
is centered at the origin. Without any loss of generality we shall prove
\eqref{mass} only for the case $i=0$.
Let us fix $l$ and $\epsilon$ as in the statement of Proposition~\ref{Prop1}.
For each $j$ with $\|j\|>rl$ we use \eqref{basic} and estimate:
\begin{equation}
\label{covdecomp}
\begin{split}
{\Bbb C}\text{ov}\big( \phi_0 ;\phi_j\big)~\leq~
&\sum_{A~\text{is}~(r,\epsilon )-\text{clean}}\rho\left( A\right)\\
&+~\sum_{A~\text{is}~(r,\epsilon )-\text{dirty}}
\rho\left( A\right)\max_{\phi}{\cal E}_{0,\phi}^A\int_0^{\tau_A}
{\Bbb I}_{\{X(s)=j\}}\dr s .
\end{split}
\end{equation}
The first term in \eqref{covdecomp} has been just estimated in
Proposition~\ref{Prop1}. Let us use $\tau_{rl}$ to denote the exit time from
${\bf B} (0,rl)$. The second term in \eqref{covdecomp} could be further
bounded above as
\begin{equation}
\label{second}
\max_{A~\text{is}~(r,\epsilon)-\text{dirty}}\max_{\phi}
{\cal E}^A_{0,\phi} \big(\tau_A >\tau_{rl}\big)
\sum_B\rho(B)\max_{\psi}{\cal E}^B_{j,\psi}\int_0^{\tau_B}
{\Bbb I}_{\{X(s)=j\}}\dr s.
\end{equation}
It is convenient to estimate the above expression in a complete generality of
time dependent random walks with bounded jump rates $a(i,j;t)$:
Let $X (t)$ be the time-inhomogeneous Markov process with the transition rates
as in \eqref{jumprates}. It is always possible to homogenize it, and to
consider
$$
\tilde{X} (t)~\df~(X(t) ,t ) .
$$
We shall use $\tilde{\E}_{(i,t)}$ to denote the law of $\tilde{X}$ with the
space-time starting point $(i,t)\in\Ztwo\times\R$.
The ${\bf B}(0,rl)$ box is decomposed to the disjoint union of sub-blocks on
the $l$-scale as:
$$
{\bf B}(0,rl)~=~\cup_{x\in {\bf B}^l (0,r)}{\bf B}(x ,l) .
$$
To a generic point $i\in\ {\bf B} (0,rl)$ we associate an $l$-block ${\bf B}_l
(i)$ according to the following rule:
$$
{\bf B}_l (i)~=~{\bf B}(x ,l)\ \ \text{if}\ i\in {\bf B}(x ,l)\
\text{for some}\ x\in\Ztwol .
$$
Given a $(r,\epsilon )$-dirty set $A\subset\Ztwo$, let us call a block ${\bf
B}(x ,l);\ x\in\Ztwol$, dirty if
$$
{\bf B}(x ,l)\cap A\neq\emptyset .
$$
We introduce now the following family of stopping times for the process
$\tilde{X} (t)$:
\begin{equation*}
\begin{split}
&T_1 =\inf_{t\geq 0}\{ {\bf B}_l (X(t ))\ \text{is dirty}\}. \\
&\,\\
&S_1 =\inf_{t\geq T_1}\{ {\bf B}_l (X(t))\neq {\bf B}_l
(X(T_1))\} . \\
&\,\\
&T_2 =\inf_{t\geq S_1}\{{\bf B}_l (X(t))\ \ \text{is dirty}\}\\
&............................\\
&S_n =\inf_{t\geq T_n}\{ {\bf B}_l (X(t))\neq {\bf B}_l
(X(T_n))\}. \\
&........................... \\
\end{split}
\end{equation*}
The condition of $A$ being $(r,\epsilon )$-dirty is readily translatable under
$\P_A$ to the sure event
$$
\left\{\tau_{rl}~>~T_{\epsilon r}\right\} .
$$
Consequently, if, as before, we use $\tau_A$ to denote the hitting time of the
set $A$ ,
$$
\tilde{\P}_{(0,0)} (\tau_A >\tau_{rl})~\leq~\tilde{\P}_{(0,0)}
(\tau_A > T_{\epsilon r} )~=~
\tilde{\E}_{(0,0)}\tilde{\E}_{\tilde{X} (T_1 )} {\Bbb I}_{\tau_A >
S_1}...
\tilde{\E}_{\tilde{X}(T_{\epsilon r})}{\Bbb I}_{\tau_A >
S_{\epsilon r}} .
$$
We claim that each of the $\epsilon r$ terms in the above product admits an
upper bound of the form
\begin{equation}
\label{dirtybound}
1~-~\left(\frac1{3c_V^2 +1}\right)^{2l} .
\end{equation}
uniformly in all Markov chains with bounded rates condition \eqref{jumprates}
and (which is the same) in all possible values of above stopping times.
Indeed let ${\bf B}_l$ be a box of side length $l$, and $i,k\in {\bf B}_l$ .
Then one strategy for a random walk starting at $i$ to hit $k$ before leaving
${\bf B}_l$ is to march to $k$ directly along some prescribed unambiguous
trajectory, say first horizontally and then vertically. Clearly if one pulls
down the rates along such a trajectory to the minimum value $1/c_V$ and pushes
the rates leading out of this trajectory to the maximal value $c_V$, then the
probability to follow the trajectory itself only decreases, but to an exactly
computable value
$$
\left(\frac1{3c_V^2 +1}\right)^{\|i-k\|} ,
$$
where the power $\|i-k\|$, of course, corresponds to the number of steps along
the trajectory. Hence \eqref{dirtybound}.
As a result:
\begin{prop}
\label{Prop2}
Uniformly in $r$ and in $(r,\epsilon)$-dirty sets $A$,
$$
\max_{\phi} {\cal E}_{0,\phi}^A \left(\tau_A >\tau_{rl}\right)
~\leq~\e^{-c_3 rl} .
$$
\end{prop}
\qed
\vskip 0.1in
\noindent
Finally,
\begin{equation}
\label{selfterm}
\begin{split}
\sum_B\rho(B)\max_{\phi}&{\cal E}^B_{j,\phi}\int_0^{\tau_B}
{\Bbb I}_{\{X(s)=j\}}\dr s\\
&=~\sum_{k=1}^{\infty}\sum_{B:\text{d}(j,B)=k}
\rho(B)\max_{\phi}{\cal E}^B_{j,\phi}\int_0^{\tau_B}
{\Bbb I}_{\{X(s)=j\}}\dr s ,
\end{split}
\end{equation}
where $\text{d}(j,B)\df\inf\{ \| j-i\| :\ i\in B\}$.
Proceeding as in the proof of Proposition~\ref{Prop2}, we readily obtain that
there exists a number $M =M (c_V ) <\infty$, such that;
$$
\max_{\phi}{\cal E}^B_{j,\phi}\int_0^{\tau_B}
{\Bbb I}_{\{X(s)=j\}}\dr s~\leq~M^k,
$$
whenever $\text{d} (j,B) =k$. On the other hand, by Theorem~\ref{Main},
$$
\sum_{B:\,\text{d}(j,B)=k}\rho (B)~\leq~\e^{-\nu k^2} ,
$$
as soon as $k>R$. Therefore, the sum in \eqref{selfterm} converges, and the
proof of Lemma~\ref{Lemma1} is, thereby, concluded
\qed
\vskip 0.2in
\noindent
{\it Proof of Theorem~\ref{Main}:}\
Let us start by introducing some additional notation: Given a finite set
$B\subset\Ztwo$ with the decomposition \eqref{Bdecomp} into the disjoint union
of
connected components $B_1,...,B_n$ we say that another set $A$ is a dry
neighbour of $B$; $A\in{\cal D}_B$, if
$$
A\cap B=\emptyset\qquad\text{but}\qquad D\cup\partial
B_l\neq\emptyset
;\ l=1,...,n.
$$
\begin{prop}
\label{Prop3}
There exists a constant $c_4 =c_4 (J)$, such that for every finite
$B\subset\Ztwo$,
\begin{equation}
\label{dryneighbour}
\sum_{A\in{\cal D}_B}\rho (A)~\leq~\e^{-c_4 |B|} .
\end{equation}
\end{prop}
The proof of Proposition~\ref{Prop3} relies on the following two basic
estimates which have been proven in \cite{DV}:
\begin{enumerate}
\item There exists a number $M=M(J)$ and a constant $c_5 =c_5 (J)$, such that,
\begin{equation}
\label{Yvan1}
\inf_{A\in{\cal D}_B}\sum_{C\subset B}\e^{J|C|}\frac{{\bf Z}(A\cup
C)}
{{\bf Z}(A)}~\geq~\e^{c_5 |B|} ,
\end{equation}
whenever $B$ is connected and $\text{diam}(B)\geq M$.
\item Let $A\neq\emptyset$ and $i\in\Ztwo\setminus A$. Then,
\begin{equation}
\label{Yvan2}
\frac{{\bf Z}(A\cup\{i\})}{{\bf Z}(A)}~\geq~\frac{c_6 (J)}{\sqrt{\dr
(i,A)}} .
\end{equation}
\end{enumerate}
The above estimates are linked to the claim of Proposition~\ref{Prop3} in the
following way:
$$
\sum_{A\in{\cal D}_B}\rho (A)
~\leq~\left(\inf_{A\in{\cal D}_B}\sum_{C_1\subset
B_1}...\sum_{C_n\subset
B_n}\frac{{\bf Z}(A\cup_1^nC_l)}{{\bf
Z}(A)}\e^{J\sum_1^n|C_l|}\right)^{-1}.
$$
If, for some $m\in [1,...,n-1]$, we regroup $B$ as
$$
B~=~B^+\cup B^-~\df~\left\{ B_1
,...,B_m\right\}\bigcup\left\{B_{m+1},...,
B_n\right\} ,
$$
then, since $A\cup_1^m C_l$ always belongs to ${\cal D}_{\cup_{m+1}^n B_l}$,
we obtain the following decoupling estimate:
\begin{equation}
\label{decouple}
\begin{split}
\inf_{A\in{\cal D}_B}&\sum_{C_1\subset
B_1}...\sum_{C_n\subset
B_n}\frac{{\bf Z}(A\cup_1^nC_l)}{{\bf
Z}(A)}\e^{J\sum_1^n|C_l|}\\
&\geq~\inf_{A\in{\cal D}_B^+}\sum_{C_1\subset
B_1}...\sum_{C_m\subset
B_m}\frac{{\bf Z}(A\cup_1^m C_l)}{{\bf
Z}(A)}\e^{J\sum_1^m|C_l|}\\
&\qquad\times\inf_{A\in{\cal D}_B^-}\sum_{C_{m+1}\subset
B_{m+1}}...\sum_{C_n\subset
B_n}\frac{{\bf Z}(A\cup_{m+1}^nC_l)}{{\bf
Z}(A)}\e^{J\sum_{m+1}^n|C_l|} .
\end{split}
\end{equation}
In particular, the claim \eqref{dryneighbour} directly follows from the
estimate \eqref{Yvan1} whenever $\text{diam}(B_l ) >M$ for each $l=1,...,n$. In
fact, in view of \eqref{Yvan1} and \eqref{decouple}, it remains to study only
the case when all connected components of $B$ are small; $\text{diam}(B_l )
0:\ B_2^{(k)}\cap (A\cup B_1^{(k_1)})
=\emptyset\} \\
&~\cdot\\
&~\cdot\\
&~\cdot\\
&~k_n~=~\max \big\{ k>0:\ B_n^{(k)}\cap (A\cup_1^{n-1} B_l^{(k_l)})
=\emptyset\} \\
\end{split}
\end{equation*}
with the convention that the maximum over an empty set equals zero.
\item For any $B$-admissible tuple $\underline{k}=(k_1,...,k_n)$;
$$
\left| B^{(\underline{k} )}\right|~\geq~|B| +\sum_1^n k_l .
$$
This follows directly from the definition of the $B$-admissibility.
\end{enumerate}
Using Proposition~\ref{Prop3} we, thereby, obtain:
\begin{equation*}
\begin{split}
\sum_{A\cap B =\emptyset}\rho
(A)~&=~\sum_{B-\text{admissible}\,\underline{k}}
\sum_{A\in{\cal D}_B^{(\underline{k})}}\rho (A)\\
&\leq~\sum_{B-\text{admissible}\,\underline{k}}\e^{-c_4 (|B|+\sum
k_l)}\\
&\leq~\e^{-c_4 |B|}\left( 1- \e^{-c_4}\right)^{-n} .
\end{split}
\end{equation*}
By the assumption \eqref{diam}, $n\leq |B|/R$. Thus it remains to choose $R
=R(J)$ so large that,
$$
\nu (J)~\df~c_4 (J) +\frac{\log (1-\e^{-c_4 (J)})}{R}~>~0,
$$
and \eqref{mainbound} follows.
\qed
\begin{thebibliography}{100}
\bibitem[BB]{BB} E. Bolthausen, D. Brydges (1998),
{\em Gaussian Surface Pinned by a Weak Potential}, preprint.
\bibitem[DGI]{DGI}
J.-D. Deuschel, G. Giacomin, D. Ioffe (1998),
{\em Concentration results for a class of effective interface
models},
preprint.
\bibitem[DV]{DV} J-D.~Deuschel, Y.~Velenik (1998), {\em Non-Gaussian
surface pinned by a weak potential}, preprint.
\bibitem[DMRR]{DMRR} F. Dunlop, J. Magnen, V. Rivasseau, P. Roche
(1992),
{\em Pinning of an Interface by a Weak Potential}, J.Stat.Phys. 87,
275-312.
\bibitem[HS]{HS} B.~Helffer and J.~Sj\"ostrand (1994),
{\em On the correlation for {Kac--like}
models in the convex case},
J.Stat.Phys. 74, 349-409 .
\end{thebibliography}
\end{document}