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Directed polymers, martingales, random 
environment
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% Directed Polymers in Random Environment: 
% Path Localization and  Strong Disorder
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\begin{document}
%\baselineskip=24pt 

\bcenter 

\large{\bf Directed Polymers in Random Environment: 
Path Localization and  Strong Disorder}


\vvs

\vvs \normalsize 


\noindent Francis COMETS
%%%\footnote{Corresponding author}
\\

\vs \small
Universit{\'e} Paris 7, \\
Math{\'e}matiques, Case 7012\\
 2 place Jussieu,
75251 Paris, France \\
email: comets@math.jussieu.fr \\

\vvs \normalsize

\noindent Tokuzo SHIGA\\

\vs \small 
Tokyo Institute of Technology\\
 Oh-okayama, Meguroku, \\
Tokyo 152-8551 Japan \\
email tshiga@math.titech.ac.jp 

\vvs \normalsize

\noindent Nobuo YOSHIDA\\

\vs \small 
Division of Mathematics \\
Graduate School of Science \\
Kyoto University,\\
Kyoto 606-8502, Japan.\\
email: nobuo@kusm.kyoto-u.ac.jp\\

\ecenter
\normalsize 
\begin{center}
January 23, 2002
\end{center}

\begin{abstract}


We consider directed polymers in random  environment. 
Under mild assumptions on the environment, we prove here:
(i) equivalence of decay rate of the partition 
function with some natural localization properties 
of the path, (ii) quantitative estimates of the decay 
of the partition function  in dimensions one or two, 
or at sufficiently low temperature,
(iii) existence of quenched free energy.
In particular, we generalize to general environments,
some of the results recently obtained by 
P. Carmona and Y. Hu for a Gaussian environment. 
We do not discuss here superdiffusivity or critical exponents.

\end{abstract}
\vspace{1cm}
\footnotesize 
\noindent{\bf Short Title.} Directed Polymers in Random Environment
 
\noindent{\bf Key words and phrases.} Directed polymers, martingales, random
environment

 
\noindent{\bf AMS 1991 subject classifications.} Primary 60K35;
secondary 60G42, 82A51, 82D30         

\normalsize
\newpage

%%%%%%%%%%%%%%%%%%%%
%
%\footnotesize 
%
%\tableofcontents
%\vspace{1cm}
%\normalsize
%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%
\SSC{Introduction and Main Results}



\subsection{
Directed Polymers in Random Environment}
%%%%%%%%%%%%%%%%%%%%%

The models we consider in this paper are defined in
terms of a random walk and of a random environment, that we introduce now:
\bitemize
\item
{\it The random walk:} 
$(\{ S_n\}_{n \geq 0}, \{P^x\}_{x \in \zd})$ is a simple random walk on 
$d$-dimensional integer lattice $\zd$.
More precisely, let $\W$ be the path space 
 $\W=\{\w=(\w_n)_{n \geq  0}; \w_n \in \zd
%%$ and $\|\w_{n+1}\!-\!\w_n\|_1=1$ for all $
, n \geq 0\}$, let  $\cF$ be the cylindric $\sigma$-field on $\W$, and, for
all $ n \geq 0$, $S_n:
\w \mapsto \w_n$ the projection map. For all $x \in \zd$ we consider 
the unique
probability measure $P^x$   on $(\W, \cF)$ such that
$S_1-S_0, \ldots, S_n-S_{n-1}$ are independent and 
$$
P^x\{ S_0=x\}=1, \; \; \; 
P^x\{ S_n\!-\!S_{n-1}=\pm \del_j\}=(2d)^{-1},\; \; \; 
j=1,2,\ldots, d, 
$$
where $\del_j =(\del_{kj})^d_{k=1}$ is the $j$-th vector of the
canonical basis of $\zd$. For  $x=0$ we will write shortly $P=P^0$.
\item 
{\it The random environment:}  
$\xi =\{\xi (x,n) : x \in \zd,\; n \ge 1  \}$ is a real,
non-constant, i.i.d. 
sequence of 
random variables defined on a probability space $(\Xi, \cE, Q)$ such that 
\bdnl{expint}
Q[\exp (\b \xi (x,n))] <\8 \; \; \; \mbox{for all $\b \in \R$.}
\edn
(All through, $Q[Y]$ denotes the $Q$-expectation of a r.v. $Y$.)
Let  $\lm (\b)$ be 
the logarithmic moment generating function of $\xi (x,n)$,
\bdnl{lm(beta)}
\lm (\b)=\ln Q[\exp (\b \xi (x,n))], \; \; \; \b \in \R. 
\edn
\eitemize
%%%%%%%%%
For all $n>0$ define the probability measure $\m_n$ on the path space 
$(\W, \cF)$ 
\bdnl{mnen}
\m_n (d\w )=P[e_n: d\w]/P[e_n],
\edn
where 
\bdnl{zetan}
e_n=e_n (\xi, S)=\exp \lef(\sum_{1 \le j \le n}
(\b \xi (S_j,j)-\lm (\b ) )\rig)
\edn
with a parameter $\b \in \R$. Here, the graph $\{ (S_j,j)\}_{j \geq 0}$ 
may be interpreted as a polymer 
chain living in the $(d+1)$-dimensional space, constrained to stretch in the
$(d+1)$-th direction, and governed by the Hamiltonian
$$ 
-\sum_{j \geq 1}
(\b \xi (S_j,j)-\lm (\b ) )\;,
$$ 
i.e. the so-called directed polymer in the environmnent $\xi$.
If $\b>0$, then the parameter $\b>0$ plays the role of temperature inverse
 in this interpretation. 
Since this Hamiltonian is parametrized by $\xi$, the polymer measure
$\m_n $ is random. Here are two standard choices for $\xi$.


%%%%%%%%
\Example{Gauss}{\it Gaussian environment} (Carmona and Hu, 2001)
%%\cite{CaHu01};  
%%%%%%
This is the case in which the distribution of $\xi (x,n)$ is 
given by standard normal distribution, so that 
%%\bdnl{lmGau}
$$
\lm (\b )=\half \b^2.
%%\edn 
$$
%%%%%%%%%
\end{example}
%%%%%%%%%
%%%%%%%%
\Example{Ber}{\it Bernoulli environment} (Bolthausen 1989, Imbrie and
Spencer 1988, Song and Zhou 1996):
%%\cite{Bol89,ImSp88,SoZh96}; 
%%%%%%
This is the case in which $\xi (x,n)$ takes 
two different values $a$ and $b$ with probability $p>0$ and $1-p>0$, 
respectively, 
so that 
%%\bdnl{lmBer}
$$
\lm (\b )=\ln (pe^{\b a}+(1\!-\!p)e^{\b b}).
%%\edn
$$
%Directed percolation can be thought as the case of 
%$0=a >b$ and zero-temperature ($\b \to \infty$) 
%%\cite{Joh00}, which however 
%is outside the scope of this paper. 
As discussed by Johansson (2000, Remark 1.8), directed percolation can be thought
as the case of $0=a >b$ and zero-temperature ($\b \to \infty$), which however  
is outside the scope of this paper.
%%%%%%%%%
\end{example}
%%%%%%%%%%%%%%%%%


We are interested in the large time behavior of the path 
$\{ S_k\}_{k=1}^n $ under the (sequence of) polymer measure 
$\m_n$. As is the case in many other models in statistical mechanics, 
one of the fundamental question to be asked is the asymptotic behavior of the 
partition function
\bdnl{Zn}
Z_n=Z_n (\xi )=P[e_n]\;.
\edn
Since $Z_n$ is a positive martingale on 
$(\Xi, \cE, Q)$, the following limit exists 
$Q$-a.s.:
\bdnl{Z8}
Z_\8 \st{\rm def.}{=} \limn Z_n\;. 
\edn 
The event $\{ Z_\8=0\}$ is measurable with respect to the tail $\s$-field
$$
\bigcap_{n \ge 1}\s [ \xi (x,j) \; ; \; j \ge n, \; x \in \zd ]
$$ 
and therefore by Kolmogorov's 0-1 law 
\bdnl{0-1}
Q \{ Z_\8 =0 \}=\mbox{0 or 1}.
\edn 
We refer to the former case as 
{\bf weak disorder}
and the latter 
as 
{\bf strong disorder}. 
It is known
%\cite{SoZh96} 
(e.g., Song and  Zhou 1996)
that for $d \ge 3$,
\bdnl{dge3}
Q \{ Z_\8 =0 \}=0 \; \; 
\mbox{if $\lm (2 \b) - 2\lm ( \b) < -\ln (1-q)$} 
\edn 
where $q=P\{ \mbox{$S_n \neq 0$ for all $n \ge 1$}\}$;
%%%See also \cite{Bol89}, \cite{Sin95} for similar results and weak disorder.
Similar results  for weak disorder were obtained by
Bolthausen (1989) and Sinai (1995).
Note that the condition in (\ref{dge3}) does hold for
small $\beta$. In dimension $d \geq 3$, this condition amounts 
to $L^2$-convergence
in (\ref{Z8}), and it allows using  the so-called second
moment method: For small $\beta$ and $d \geq 3$, 
%the polymer is diffusive
%\cite{ImSp88}, \cite{Bol89}, and the invariance principle holds
%for almost every environment \cite{AlZh96}.
Imbrie and Spencer (1988) first, then Bolthausen (1989) with
martimgales techniques, proved that the polymer
is diffusive; more recently Albeverio and  Zhou (1996) 
showed that the invariance principle holds
for almost every environment.
On the other hand,  for strong disorder, it can be seen that 
\bdnl{KaPe}
Q \{ Z_\8 =0 \}=1 \; \; \mbox{if $\b \lm^\pri (\b )-\lm (\b ) \ge \ln (2d)$.}
\edn 
This was shown 
%in \cite{KaPe76} 
by Kahane and Peyri{\`e}re (1976) for a different model 
called Mandelbrot martingale (or, equivalently, multiplicative chaos), 
where graphs  $\{ (S_j,j)\}_{j \geq 0}$ are replaced 
by infinite paths, without loops and starting
from the root, on the $d$-ary tree. Although the directed polymer we 
are considering here is more
intricate due to correlations, the same argument applies 
as far as to deduce (\ref{KaPe}). 
Recently, P. Carmona and Y. Hu 
(2001)
%%\cite{CaHu01} 
proved for Gaussian environment
that for all $\beta$,
\bdnl{d<3}
Q \{ Z_\8 =0 \}=1,\; \; \; d=1,2,
\edn
which, together with (\ref{KaPe}), displays a non-trivial dependence 
on the dimension.

\medskip

In the present paper, we consider general environment and 
present some results for strong disorder case: $Q \{ Z_\8 =0 \}=1$, 
including the extension of (\ref{d<3}) to non-Gaussian case. 
Using martingale analysis, we also
obtain natural localization properties which 
characterize the strong disorder regime. More precisely, decay of the
partition function is equivalent to concentration of the path on
favourite sites. 
Though, we
will not discuss here superdiffusivity or critical exponents, we refer
to
Johansson (2000), Licea {\it et al} (1996), Petermann (2000) and Piza
(1997) for
%%\cite{Joh00}, \cite{LiNePi96}, \cite{Pet00}, \cite{Piz97} for 
rigourous results in this direction.
%%%%%%%%%%%%%%%%%%
\subsection{Results}
%%%%%%%%%%%%%%%%%%%%%
On the product space $(\W^2, \cF^{\otimes 2})$, we consider
the probability measure $\m_{n}^{\otimes 2}=\m_{n}^{\otimes 2}(d\w,
d\tw)$, 
that we will view as the distribution of the couple $(S, \tl{S})$ with 
 $\tl{S}=\{ \tl{S}_k\}_{k \ge 0}$  an independent copy 
of $S=\{ S_k\}_{k \ge 0}$ with law $\m_n$.
An important role in the analysis is played by the random sequence
\bdnl{Bn}
I_n=\m_{n-1}^{\otimes 2}(S_n =\tl{S}_n)\;,
\edn
which conveys some information on the localization of paths under
$\m_n$, see (\ref{<local}) below.
Our basic result relates the partition function $Z_n$  and the
expected intersection time $\sum_{j \leq n} I_j$ of two independent
polymers in the same environment.

%%%%%%%%%%%%%%%%%%%
\Theorem{equiv}
%%%%%%%%%%%%%%%%%%
Let $\b \neq 0$. Then, 
\bdnl{equiv1}
\{ Z_\8 =0 \} = \lef\{ \sum_{n \ge 1}I_n =\8 \rig\}, \; \; \; 
\mbox{$Q$-a.s.}.
\edn 
%Moreover, there exist $c_1,c_2 \in (0,\8)$ such that
%\bdnl{equiv2}
%\{ Z_\8 =0 \} \sub 
%\lef\{ -c_1\ln Z_n \le \sum_{1 \le j \le n}I_j 
%\le -c_2\ln Z_n 
%\; \; \mbox{for large enough $n$'s}\rig\}
%, \; \; \; 
%\mbox{$Q$-a.s..}
%\edn 
Moreover, if $Q\{ Z_\8 =0 \}=1$, then 
there exist $c_1,c_2 \in (0,\8)$ such that
\bdnl{equiv2}
 -c_1\ln Z_n \le \sum_{1 \le j \le n}I_j 
\le -c_2\ln Z_n 
\qquad \mbox{for large enough $n$'s}
, \; \; \; 
\mbox{$Q$-a.s..}
\edn 



%%%%%%%%%%%%%%%
\end{theorem}
%%%%%%%%%%%%
As a consequence we see that fast decay property of $Z_n$ 
is reflected in some specific localization property of the 
path $\{S_n\}_{n \ge 1}$ under the random measure (\ref{mnen}). 
Our main interest in the following corollary is the case of 
$a_n=cn$, $n \geq 1$ for a positive constant $c>0$. 
%%%%%%%%%%%%%%%%%%%%%
\Corollary{exdec}
%%%%%%%%%%%%%%%%%
For $\b \neq 0$ and a sequence $a_n \nearrow \8$ of positive numbers, 
the following properties are equivalent:
\bds
\item[(Z1)]
There exists $c>0$ such that 
\bdnl{Zexdec}
Q \lef\{ 
\inflim_{n \nearrow \8}  -\frac{1}{a_n}\ln Z_n \ge c \rig\}=1.
\edn
\item[(I1)]
There exists $c>0$ such that 
\bdnl{I1}
Q \lef\{ 
\inflim_{n \nearrow \8} 
\frac{1}{a_n}\sum_{1 \le j \le n}I_j\ge c
\rig\}=1.
\edn 
\eds
%%%%%%%%%%%%%%%
\end{corollary}
%%%%%%%%%%%%
%%%%%%%%%%%%%%
\Remark{CaHu1.2}
%%%%%%%%%%%
The equivalence presented in 
\Thm{equiv} was 
%previously  
shown first
by Carmona and Hu 
%%%\cite[Theorem 1.1, Proposition 5.1]{CaHu01}
(2001,  Theorem 1.1 and Proposition 5.1)
in the Gaussian case. 
%%%%%%%%
\end{remark}
%%%%%%%%
%%%%%%%%%
\Proposition{max}
%%%%%%%%%
(i) Assume  that Property (Z1) in \Cor{exdec} holds with $a_n=n$. Then, 
there is a constant $c \in (0,\8)$ such that 
\bdnl{max}
\suplim_{n \nearrow \8}I_n
\ge c\; \; \; \mbox{$Q$-a.s..}
\edn 
(ii) If 
%%%%%%on the contrary, 
$Q\{Z_{\8}>0 \}=1 $, then 
$$
\lim_{n \nearrow \8}I_n =0,
\quad \mbox{$Q$-a.s.}
$$
Assume now that 
the (stronger) condition in (\ref{dge3}) holds, i.e.
$\lm (2 \b) - 2\lm ( \b) < -\ln (1-q)$. Then, 
there is a constant $c >0$ such that 
\bdnl{max0}
I_n = O(n^{-c})
\edn 
in $Q$-probability.
%%%%%%%%%%%%%%%
\end{proposition}
%%%%%%%%%%%%%%%%%%%
\Remark{favorite}
%%%%%%%%%%%%%%%%%%%%
A natural quantity of interest here, relating to localization phenomenon, 
is the favorite site for the path at time $n$.
First observe that 
\bdnl{<local}
\max_{x \in \zd} \m_{n-1}(S_n=x)^2 \leq I_n \leq \max_{x \in \zd}
\m_{n-1}(S_n=x)
\;.
\edn 
Therefore, all the statement we obtained for $I_n$ 
can be translated into those for 
$\max_{x \in \zd}\m_{n-1}(S_n=x)$. 
In particular, we showed in 
\Prop{max} that the probability of the favorite site vanishes for weak
disorder, but does not vanish for strong disorder.
In the latter case the polymer 
localizes (in a set of lattice points depending on the evironment), 
though in the former it spreads out somewhat alike the usual
simple random walk.
%%%%%%%%%%%
\end{remark}
%%%%%%%%%%%%%%%
Some sufficient conditions for (Z1) and (I1)
in \Cor{exdec} are provided by  
the following result
%%%%%%%%%%%%%%%%%%%
\Theorem{extinct}
%%%%%%%%%%%%%%%%%%
Consider the following property:
\bds
\item[(Z2)]
There exist constants $c \in (0,\8)$ and $\tht \in (0,1)$ such that 
\bdnl{Znp}
Q[Z_n^\tht ]  \leq c\exp (-a_n) \; \; \; n \ge 1.
\edn
\eds
Then, 
\bds
\item[(a)]
(Z2) implies (Z1) in \Cor{exdec} if
%%\bdnl{sum=8}
$$
\sum_{n \ge 1}\exp (-\del a_n)<\8 \; \; \; 
\mbox{for some $\del \in (0,1)$.}
%%\edn 
$$
\item[(b)]
(Z2)  holds for $a_n =cn$ for some $c>0$ if 
\bdnl{(1+w)log(1+w)} 
\b \lm^\pri (\b )-\lm (\b ) >\ln (2d).
\edn 
\item[(c)]
(Z2) holds for $d=1,2$ with 
\bdnl{d=1,2}
a_n =\lef\{ \barray{ll} c_1 n^{1/3} 
& \mbox{if $d=1$} \\
c_2 \sqrt{\ln n} 
& \mbox{if $d=2$}
\earray \rig.
\edn
where $c_1, c_2 \in (0,\8) $ are some constants. 
In particular,  
for $Q$-a.s.,
$$
%%\bdnl{dle2}
Z_n \lef\{ \barray{ll} 
=\cO \lef( \exp (-\frac{c_1 }{2}n^{1/3}) \rig) 
& \mbox{as $n \nearrow \8$ if $d=1$} \\
\longrightarrow 0
& \mbox{as $n \nearrow \8$ if $d=2$.}
\earray \rig.
%%\edn
$$
\eds 
%%%%%%%%%%%%%%%
\end{theorem}
%%%%%%%%%%%%%%
%%%%%%%%%%%%%%
\Remark{CaHu1.1}
%%%%%%%%%%%
\Thm{extinct}(c) generalizes 
%%%\cite[Theorem 1.1]{CaHu01} 
Theorem 1.1 in Carmona and Hu (2001) to
non-gaussian environments.
%where 
%the case of Gaussian environment is obtained.
%%(however without convergence rate).
Moreover, we give here a quantitative bound for the rate of decay. 
%%%%%%%%
\end{remark}
%%%%%%%%
Finally, we remark that the ``quenched free energy''
$$
\limn\frac{1}{n}\ln Z_n 
$$
exists $Q$-a.s. under our assumption (\ref{expint}). 

\Proposition{press}
%%%%%%%%%%%%%%%%%%
The limit 
%%\bdnl{psi(beta)}
$$
\psi (\b )=\limn \frac{1}{n} Q[\ln Z_n] \in (-\8, 0]
%%\edn
$$ 
exists. 
Moreover, 
for any $\e >0$, there is an $n_0=n_0( \b, \e) < \8$ such that 
\bdnl{concent}
Q \lef\{  \lef| \frac{1}{n}\ln Z_n  -\psi (\b )\rig| >\e \rig\} 
\le \exp \lef( -\frac{\e^{2/3}n^{1/3} }{4 } \rig) \;,
\; \; \; n \ge n_0.
\edn
As a consequence, 
%%\bdnl{press}
$$
\limn \frac{1}{n}\ln Z_n =\psi (\b ) , \; \; \; 
\mbox{$Q$-a.s.}.
%%\edn
$$
%%%%%%%%%%%%%%%
\end{proposition}
%%%%%%%%%%%%%%
%%%%%%%%%%%%
\Remark{press}
%%%%%%%%
The inequality (\ref{concent}) is a concentration 
inequality with the streched exponential decay rate. 
An inspection of our proof reveals that an exponential 
concentration can be obtained by a slightly stronger assumption.
In fact, if we assume that
there is 
$\del >0$ such that 
\bdnl{exp2}
Q \left[ \exp (\del |\xi (x,n)|^2) \right] <\8, 
\edn 
then, we obtain the following;  
for any $\e >0$, there is an $n_0=n_0( \b, \e) < \8$ such that 
\bdnl{concent2}
Q \lef\{  \lef| \frac{1}{n}\ln Z_n  -\psi (\b )\rig| >\e \rig\} 
\le \exp \lef( -\frac{\e^{2}n }{c} \rig) \;,
\; \; \; n \ge n_0.
\edn 
where $c=c(\b)>0$. See \Rem{press2} below.
%%%%%%%%%%%%
\end{remark}
%%%%%%%%%%%
%%%%%%%%%%%%
\Remark{MC}
%%%%%%%%
We can define a similar model by considering 
a Markov chain $(\{ S_n\}_{n \geq 0}, \{P^x\}_{x \in \Gm})$ 
on a certain state space $\Gm$ instead of the random walk on $\zd$. 
The proofs presented in this paper apply without change to 
this generalization. 
%%%%%%%%%%%%
\end{remark}
%%%%%%%%%%%
%%%%%%%%%%%%
\SSC{Proof of \Thm{equiv}}
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
We begin by proving the following general estimate. 
%%%%%%%%%%%
\Lemma{gCH}
%%%%%%%%%
Let $X_i$, $1 \le i \le m$ 
be non-constant, square integrable i.i.d. random variables on a 
probability space $(\Xi, \cE, Q)$ such that 
$$
Q[\exp (X_1)]=1,\; \; \; Q[\exp (4X_1)] <\8.
$$
For a probability 
distribution $\{\a_i \}_{1 \le i \le m}$ on $\{ 1,\ldots, m \}$, 
define a centered random variable $U >-1$ by 
$$
U=\sum_{1 \le i \le m}\a_i \exp (X_i)-1.
$$ 
Then, there exists a constant $c \in (0,\8)$, independent 
of $\{\a_i \}_{1 \le i \le m}$ such that 
\bdmn 
\frac{1}{c} \sum_{1 \le i \le m}\a_i^2 & \le &
Q \left[ \frac{U^2}{2+U} \right], \label{gCH0} \\
\frac{1}{c} \sum_{1 \le i \le m}\a_i^2 & \le &
-Q\left[\ln (1+U)  \right]  \le 
c \sum_{1 \le i \le m}\a_i^2, \label{gCH} \\
&  & Q\left[\ln^2 (1+U)  \right]   \le  
c \sum_{1 \le i \le m}\a_i^2. \label{gCH2}
\edmn 
%%%%%%
\end{lemma}
%%%%%%%%%%
\Remark{CaHu2.2}
%%%%%%%%%%
These estimates are proved in 
(Carmona and Hu 2001)
%%\cite{CaHu01} 
for Gaussian 
case by runing Brownian motion and making use of It{\^o}'s formula. 
Here, we present a simple argument 
which works for general case. 
%%%%%%%%%
\end{remark}
%%%%%%%%%%%%

\noindent Proof of \Lem{gCH}:
In this proof,  we let $c_1, c_2, \ldots $ stand for constants which are 
independent of $\{\a_i \}_{1 \le i \le m}$.
We have by a standard argument 
\bdnn
Q[|U|^2] & = & c_1 \sum_{1 \le i \le m}\a_i^2, \\
Q[|U|^p] & \le & c_2 \sum_{1 \le i \le m}\a_i^2, \; \; \; 2 \le p \le 4.
\ednn
Indeed, explicit computations yield such
estimates when  $p=2,4$, and the general case follows by interpolating 
by H{\"o}lder's
inequality $Q[|U|^p] \leq Q[|U|^2]^{(4-p)/2} Q[|U|^4]^{(p-2)/2}$.
Therefore, 
\bdnn
c_1\sum_{1 \le i \le m}\a_i^2 
 & = & Q[U^2] \\
 & = & Q\lef[ \frac{|U|}{\sqrt{2+U}}|U|\sqrt{2+U} \rig] \\
 & \le & Q\lef[ \frac{U^2}{2+U} \rig]^{1/2}Q \lef[ 2U^2+|U|^3 \rig]^{1/2} \\
 & \le & c_3Q\lef[ \frac{U^2}{2+U} \rig]^{1/2}
\lef( \sum_{1 \le i \le m}\a_i^2 \rig)^{1/2},
\ednn
which proves (\ref{gCH0}). 

To prove the other inequalities, it is convenient to 
introduce a function $\vp :(-1,\8) \ra [0,\8)$ as follows;
$$
\vp (u)=u-\ln (1+u),
$$
so that  
$$
-Q \left[ \ln (1+U) \right] =  Q \left[ \vp (U) \right].
$$
Since 
$$
\frac{1}{4}\frac{u^2}{2+u} \le \vp (u), \; \; u >-1,
$$
the left-hand-side inequality of (\ref{gCH}) follows from 
(\ref{gCH0}). The right-hand-side inequality can be seen as follows.
We have for any $\e \in (0,1)$, 
\bdnn
Q  \left[ \vp (U) \right]
& = & 
Q [\vp (U): 1+U \ge \e ]+Q [\vp (U): 1+U \le \e ] \\
& \le & 
Q [\vp (U): 1+U \ge \e ]-Q [\ln (1+U): 1+U \le \e ].
\ednn 
Since $\vp \leq \half (u/\e)^2 $ if $1+ u \ge \e$, 
\bdmn
Q [\vp (U): 1+U \ge \e ] 
& \le & \half \e^{-2}Q [U^2] \nn \\
& = & \half \e^{-2}c_1\sum_{1 \le i \le m}\a_i^2. 
\label{1+U>} \edmn
We now set $\gm=-Q[X_1] \ge 0$ and choose 
$\e >0$ so small that $\ln (1/\e)-\gm \ge 1$. We introduce another 
centered random variable 
$$
V=\sum_{1 \le i \le m}\a_i (X_i+\gm). 
$$
We then see from  
Jensen's inequality that 
\bdnn
\{ 1+U \le \e \} & \sub & \{ V-\gm \le \ln (1+U) \le \ln \e \} \\
& \sub & \{ -\ln (1+U) \le -V +\gm   \}\cap \{ 1 \le -V \}.
\ednn
We consequently have that
\bdnn
-Q [\ln (1+U): 1+U \le \e ] 
& \le & 
Q [-V:  1 \le -V]
+\gm Q \{ 1 \le -V \} \\
& \le & (1+\gm)Q[V^2] \\
& = & c_4\sum_{1 \le i \le m}\a_i^2. 
\ednn
This, together with (\ref{1+U>}) proves the right-hand-side 
inequality of (\ref{gCH}). The proof of (\ref{gCH2}) is similar 
as above. In fact, since 
$|\ln (1+u)| \le \e^{-1}\ln (\e^{-1})|u|$ if $\e  \le 1+u$, we have that
$$
Q [\ln^2 (1+U): \e \le 1+U ] \le  
\e^{-2}\ln^2 (\e^{-1})Q [U^2].
$$ 
We see on the other hand that
\bdnn
\{ 1+U \le \e \} & \sub & \{ V-\gm \le \ln (1+U) \le \ln \e \} \\
& \sub & \{ \ln^2 (1+U) \le 2V^2 +2\gm^2 \}\cap \{ 1 \le -V \}.
\ednn
We obtain as a consequence that
\bdnn
Q [\ln^2 (1+U): 1+U \le \e ]  &\le &  2Q[V^2]+2\gm^2Q \{ 1 \le -V \}\\
&\le & c_5\sum_{1 \le i \le m}\a_i^2. 
\ednn
%%%%%%%%%%%%
$\Box$

\vs
%%%%%%%%%%%%
To prove \Thm{equiv}, it is enough to prove the following 
(\ref{sono1}) and (\ref{sono2}):
\bdnl{sono1}
\{ Z_\8 =0 \} \sub \lef\{ \sum_{n \ge 1}I_n =\8 \rig\}, \; \; \; 
\mbox{$Q$-a.s.},
\edn
There are $c_1,c_2 \in (0,\8)$ such that
\bdnl{sono2}
\lef\{ \sum_{n \ge 1}I_n =\8 \rig\} \sub 
\lef\{ -c_1\ln Z_n \le \sum_{1 \le j \le n}I_j 
\le -c_2\ln Z_n 
\; \; \mbox{for large enough $n$'s.}\rig\}
, \; \; \; 
\mbox{$Q$-a.s..}
\edn 
The proof of (\ref{sono1}) and (\ref{sono2}) is 
based on Doob's decomposition for the process 
$-\ln Z_n$. It is convenient to introduce some more notations. 
For a sequence $(a_n )_{n \geq 0}$ (random or non-random), we set 
$\D a_n=a_n-a_{n-1}$ for $n \geq 1$. 
We denote by $\cE_n$ the $\s$-field generated  by 
$\{ \xi (x,j) \; ; \; 1 \le j \le n, \; x \in \zd \}$, and we denote by 
$Q^\xi_{n}$  the 
conditional expectation with respect to $Q$ given $\cE_n$.
%$\cE_n$ and $Q^\xi_{n}$ denote, respectively, 
%the $\s$-field generated  by 
%$\{ \xi (x,j) \; ; \; 1 \le j \le n, \; x \in \zd \}$ and the 
%conditional expectation with respect to $Q$ given $\cE_n$.
 Let us now recall Doob's decomposition in this context; 
any $(\cE_n)$-adapted process 
$X=\{ X_n \}_{n \ge 0} \sub L^1(Q)$ can be decomposed in a unique way as 
$$
X_n=M_n (X)+A_n (X),\; \; \; n \geq 1,
$$
where $M (X)$ is an $(\cE_n)$-martingale and 
$$
A_0 =0, \; \; \D A_n =Q^\xi_{n-1}[\D X_n ], \; \; \; n \ge 1.
$$
$M_n (X)$ and $A_n (X)$ are called respectively, 
the martingale part and the compensator of the process $X$.
If $X$ is a square integrable martingale, then the compensator 
$A_n (X^2)$ of the process 
$X^2=\{ (X_n)^2 \}_{n \ge 0} \sub L^1(Q)$ is denoted by $\lan X \ran_n$ 
and is given by the following formula;
$$
\D \lan X \ran_n =  Q^\xi_{n-1}[(\D X_n)^2] 
$$
Here, we are interested in the Doob's decomposition of 
$X_n=-\ln Z_n$, whose martingale part and the compensator will 
be henceforth denoted $M_n$ and $A_n$ respectively;
\bdnl{dec}
-\ln Z_n =M_n+A_n.
\edn
 To compute $M_n$ and $A_n$, 
we introduce 
$$
%%\bdn
U_n   =  \m_{n-1}[ \exp (\b \xi (S_n,n)\!-\!\lm (\b ))] -1 \;.
%%\edn
$$ 
It is then clear that 
\bdnl{Z/Z=1+U}
Z_n/Z_{n-1}=1+U_n
\edn
and hence that
\bdnl{DADB}
\D A_n=-Q^\xi_{n-1}\ln (1+U_n), \; \; \; 
\D M_n=-\ln (1+U_n)+Q^\xi_{n-1}\ln (1+U_n).
\edn 
In particular, 
\bdnl{<M><ln2}
\D \lan M \ran_n \le Q^\xi_{n-1}\ln^2 (1+U_n).
\edn
On the other hand, we have that 
$$
I_n=\sum_{|z|_1 \le n} \m_{n-1}(S_n=z)^2 .
$$ 
We now claim that there is a constant $c \in (0,\8)$ such that
\bdmn
\frac{1}{c}I_n & \le & \D A_n \le c I_n, \label{B<A<B}\\
\D \lan M \ran_n & \le & cI_n. \label{D<M><I}
\edmn
In fact, we obtain these from (\ref{DADB}), (\ref{<M><ln2}) and 
\Lem{gCH}; 
$\{ X_i \}$, $\{ \a_i \}$ and $Q$ in the lemma play roles of 
$\{ \b \xi (z,n)-\lm (\b)\}_{|z|_1 \le n}$, 
$\{ \m_{n-1}(S_n=z) \}_{|z|_1 \le n}$ and $Q^\xi_{n-1}$.

We now conclude (\ref{sono1}) from (\ref{B<A<B}), (\ref{D<M><I}) 
as follows (the equality and the inclusions here being understood 
as $Q$-a.s.);
\bdnn
\lef\{ \sum_{n \ge 1}I_n <\8 \rig\} & = & \{ A_\8 <\8 \} \\ 
& \sub & \{ A_\8 <\8, \; \lan M \ran_\8 <\8 \} \\
& \sub & \{ A_\8 <\8, \; \mbox{${\dps \limn M_n}$ exists and is finite} \} \\
& \sub & \{ Z_\8 >0\}.
\ednn 
Here, in the third line, we have used a well-known property 
of a martingale,
e.g.  (4.9) page 255 in Durrett (1995).
%%\cite[page 255, (4.9)]{Dur95}. 

Finally we prove (\ref{sono2}). By (\ref{B<A<B}), it is enough 
to show that
\bdnl{sono3}
\lef\{ A_\8 =\8 \rig\} 
\sub \lef\{ \limn-\frac{\ln Z_n}{A_n}=1 \rig\},\; \; \; \mbox{$Q$-a.s..}
\edn
Let us thus suppose that $A_\8 =\8$. 
If $\lan M \ran_\8 <\8 $, then again by 
%%%\cite[page 255, (4.9)]{Dur95},
 (4.9) page 255 in Durrett (1995),  
${\dps\limn M_n}$ exists and is finite and therefore (\ref{sono3}) holds. 
If $\lan M \ran_\8 =\8 $, then 
$$
%%\bdnl{LLNforM}
-\frac{\ln Z_n}{A_n}  
= \frac{M_n}{\lan M \ran_n} \frac{\lan M \ran_n}{A_n} +1 \to 1
\; \; \; \mbox{$Q$-a.s.}
%%\edn
$$
by (\ref{B<A<B}), (\ref{D<M><I}) and 
the law of large numbers for martingales 
%%%\cite[page 255, (4.10)]{Dur95}.
(see (4.10) page 255 in Durrett 1995).
This completes the proof. 
%%%%%%%%%
$\Box$

\vs 
%%%%%%%%%%%%%%%%%%%%%%
\SSC{Proof of \Prop{max}} \label{pmax}
%%%%%%%%%%%%%%%%%
%%%%%%%
%The statement in (i) directly follows from the upper bound in
%(\ref{<local}) and from (\ref{equiv2}).  The first statement in (ii)
%is derived using the lower bound and convergence of $I_n$ to 0. 
%%%%%%%%%
The statement in (i) directly follows from (\ref{I1}).  
The first statement in (ii)
is derived using the the convergence of $I_n$ to 0. 
We now prove (\ref{max0}). 
%%%%%%%%
Since $Q\{ Z_\8 >0 \}=1$ 
in the present case -see  (\ref{dge3})-,it is enough to show that 
\bdnl{maxp1}
 Z_{n-1}^2 I_n = O(n^{-c})
\edn
in $Q$-probability. With $\gamma= \lm(2 \b) -2 \lm (\b)< -\ln(1-q)$, 
we compute
\bdnn
%%%%%%%%%%%
%Q\lef[  (Z_{n-1} \max_{x}\m_{n-1}\{ S_n =x\})^2 \rig] & \leq & 
%%%%%%%%%%%%%%%
Q\lef[  Z_{n-1}^2I_n\rig] 
& = & Q\lef[ P^{\otimes 2} (e_{n-1} (\xi , S) e_{n-1} (\xi , \tl{S}) 
: S_n=\tl{S}_n)\rig]\\
&  = & P^{\otimes 2} ( Q[e_{n-1} (\xi , S) e_{n-1} (\xi , \tl{S}) ]
: S_n=\tl{S}_n)\\
&  = &   P^{\otimes 2} \lef( \exp \lef\{ \gamma \sum_{j=1}^{n-1} {\bf
  1}_{S_j=\tl{S}_j}
\rig\} : S_n=\tl{S}_n \rig)\\
& \leq &  P^{\otimes 2} \lef(
\exp \lef\{ \alpha \gamma \sum_{j=1}^{n-1} {\bf
  1}_{S_j=\tl{S}_j}\rig\}\rig)^{1/\alpha}  
P^{\otimes 2}( S_n=\tl{S}_n)^{1/\alpha'}\;,
\ednn
using H{\"o}lder's inequality with  conjugate exponent $\alpha,
\alpha'$. Since $\sum_{j \geq 1} 
{\bf  1}_{S_j = \tl{S}_j}$ is geometrically distributed with failure
probability $1-q \in (0,1)$ with $q$ as in (\ref{dge3}), the first
factor in the right-hand side is bounded for $\alpha \gamma < -\ln(1-q)$.
The second factor is $O(n^{-d/(2 \alpha')})$. From this we obtain 
 (\ref{max0}) for  arbitrary $c < d[1+\gamma/ \ln(1-q)]/2$.
$\Box$

%%%%%%%%%%%%
\SSC{Proof of \Thm{extinct}}
%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%
\subsection{Proof of part (a)} 
%%%%%%%%%%%%%%%% 
This follows easily by the Borel-Cantelli lemma. 
$\Box$

\vs 
%%%%%%%%%%%%%%%%%
\subsection{Proof of part (b)} 
%%%%%%%%%%%%%%%% 
For $\tht \in (0,1)$, by the subadditive estimate 
$(u+v)^\tht \leq u^\tht +v^\tht$, 
$u, v >0$, we get
$$
Z_n^\tht \leq (2d)^{-\tht} 
\sum_{x, |x|_1=1} \exp (\tht \b \xi (x,1)-\tht \lm (\b) )(Z_{1,n}^x)^\tht
$$
where 
$$
Z_{1,n}^x=P^x\exp \lef(\sum_{1 \le j \le n-1}
(\b \xi (S_j,j+1)-\lm (\b ) )\rig).
$$
Since $Z_{1,n}^x$ has the same law as $Z_{n-1}$, 
$$
Q[Z_n^\tht] \leq r (\tht )Q[Z_{n-1}^\tht],
$$
where
$$
r(\tht )=(2d)^{1-\tht }Q[\exp (\tht \b \xi (x,1)-\tht \lm (\b))]
$$
Note that $\tht \mapsto \ln r (\tht )$ is convex and that 
$\ln (2d)=\ln r(0) > \ln r (1)=0$. Therefore $ r (\tht )<1$ for some 
$\tht \in (0,1)$ if and only if 
$0<\lef. \frac{d \ln r (\tht )}{d\tht} \rig|_{\tht =0}$, 
which is equivalent to 
(\ref{(1+w)log(1+w)}). 
%%%%%%%%%
$\Box$ 

\vs
%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%
\subsection{Proof of part (c)}
%%%%%%%%%
\Lemma{Lig}
%%%%%%%%%
For $\tht \in [0,1]$ and $\Lm \sub \zd$, 
\bdnl{Lig}
Q\lef[ Z_{n-1}^\tht I_n \rig]
\ge \frac{1}{|\Lm |}
Q\lef[ Z^\tht_{n-1}\rig] -\frac{2}{|\Lm |}P(S_n \not\in \Lm)^\tht.
\edn
%%%%%%%
\end{lemma}
%%%%%%%%
Proof: 
Following the argument 
%%of \cite[page 453]{Lig85},
in Liggett (1985, page 453),
 we see that 
\bdnn
I_n
& \ge & \sum_{z \in \Lm }\m_{n-1}(S_n =z)^2 \\
& \ge & \frac{1}{|\Lm |} \m_{n-1}(S_n \in \Lm)^2 \\
& = & \frac{1}{|\Lm |} \lef( 1- \m_{n-1}(S_n \not\in \Lm) \rig)^2 \\
& \ge & \frac{1}{|\Lm |}\lef( 1- 2\m_{n-1}(S_n \not\in \Lm) \rig) \\
& \ge & \frac{1}{|\Lm |}\lef( 1- 2\m_{n-1}(S_n \not\in \Lm)^\tht \rig).
\ednn
Note also that
\bdnn 
Q\lef[ Z^\tht_{n-1}\m_{n-1}(S_n \not\in \Lm)^\tht \rig]
& \le & 
Q\lef[ Z_{n-1}\m_{n-1}(S_n \not\in \Lm) \rig]^\tht \\
& = & P(S_n \not\in \Lm)^\tht.
\ednn 
We therefore see that 
\bdnn
Q\lef[ Z_{n-1}^\tht I_n \rig]
& \ge &\frac{1}{|\Lm |}
Q\lef[ Z^\tht_{n-1}\rig]
-\frac{2}{|\Lm |}Q\lef[ Z^\tht_{n-1}\m_{n-1}(S_n \not\in \Lm)^\tht \rig] \\
& \ge &\frac{1}{|\Lm |}
Q\lef[ Z^\tht_{n-1}\rig] -\frac{2}{|\Lm |}P(S_n \not\in \Lm)^\tht.
\ednn
%%%%%%%%%
$\Box$ 

\vs
%%%%%%%%
Define a function $f : (-1, \8) \ra [0,\8 )$ by 
$$
f (u)=1+\tht u -(1+u)^\tht.
$$
It is then clear that there are constants $c_1, c_2 \in (0,\8)$ such that 
\bdnl{<f(u)<}
\frac{c_1 u^2}{2+u} \le f(u) \le c_2 u^2 \; \; \; 
\mbox{for all $u \in (-1,\8)$.}
\edn
We see from (\ref{Z/Z=1+U}), (\ref{<f(u)<}) and (\ref{gCH0}) that 
\bdmn
Q^\xi_{n-1}\D Z_n^\tht 
& = & Z_{n-1}^\tht Q^\xi_{n-1}\lef( (1+U_n)^\tht -1\rig) \nn \\
& = & -Z_{n-1}^\tht Q^\xi_{n-1}f(U_n) \nn  \\
& \le & -c_3Z_{n-1}^\tht I_n.\nn
\edmn 
We therefore have by (\ref{Lig}) that 
\bdnl{aLig}
QZ_n^\tht 
\le \lef( 1 -\frac{c_3}{|\Lm |} \rig)
Q\lef[ Z^\tht_{n-1}\rig] -\frac{2c_3}{|\Lm |}P(S_n \not\in \Lm)^\tht.
\edn
For $d=1$, set $\Lm =(-n^{2/3}, n^{2/3}]$. Then,
$$
P(S_n \not\in \Lm) = P\lef(\lef| \frac{S_n}{n^{1/2}} \rig| \ge n^{1/6} \rig)
\le 2\exp (-\frac{n^{1/3}}{2}),
$$
so that (\ref{aLig}) reads, 
$$
Q Z_n^\tht 
\le \lef(1-\frac{c_3}{2n^{2/3}}\rig)
Q\lef[ Z^\tht_{n-1}\rig] +4c_3\exp (-\frac{n^{1/3}}{2}).
$$
It is not difficult to conclude (\ref{d=1,2}) for $d=1$ from the above.  

For $d=2$, we set 
$$ \Lm =( -n^{1/2}\ln^{1/4}n, n^{1/2}\ln^{1/4}n]^2 $$ 
to get 
(\ref{d=1,2}) for $d=2$ in a similar way as above.   
%%%%%%%%%%%% 
$\Box$

\vs 
%%%%%%%%%%%%%%
%%%%%%%%%%%%

\vvs 


\vs 

%%%%%%%%%%%%
\SSC{Proof of \Prop{press}}
%%%%%%%%%%%%%%%%%%

Though the first statement 
 is well known,
we give a proof here for definiteness. Note that for $m, n \geq 1$,
$$
Z_{n+m} = Z_n \sum_x \m_n \{ S_n=x \}  Z_{n,m}^x\;,
$$
where $Z_{n,m}^x = P^x [ \exp \{ \sum_{1 \leq j \leq m} 
\beta \xi(S_j, n+j) - \lambda (\beta)\}] $ has the same law as $Z_m$. By 
Jensen's inequality, 
$$
\ln Z_{n+m}  \geq  \ln Z_n + \sum_x \m_n \{ S_n=x \}  \ln Z_{n,m}^x\;.
$$
Recall the notations 
$\cE_n$ and $Q^\xi_{n}$ introduced in the proof of Theorem \ref{Thm.equiv}.
Taking expectation and using independence, we obtain
\bdnn
Q[ \ln Z_{n+m} ] & \geq & Q[\ln Z_n] + Q[\sum_x \m_n \{ S_n=x \}  
Q^\xi_{n} [\ln Z_{n,m}^x]]\\
& = &  Q[\ln Z_n] + Q[\ln Z_m]\;,
\ednn
i.e., $Q[\ln Z_n]$ is super-additive.
From the super-additive Lemma we see that
$$
\frac{1}{n} Q[\ln Z_n] \to \psi (\b )=\sup_n \frac{1}{n} Q[\ln Z_n] \;
$$
as $n \to \8$.
\vs

To prove (\ref{concent}) we write $\ln Z_n - n \psi( \b)$ as a sum of 
$(\cE_j)_{1 \leq j \leq n}$-martingale differences,
$$
\ln Z_n - Q[ \ln Z_n ] = \sum_{j=1 }^n V_{n,j} \,
$$
with $V_{n,j} =Q^\xi_{j}[ \ln Z_n ]-Q^\xi_{j-1}[ \ln Z_n ]$. 
Set
$$
e_{n,j}  = 
\exp \lef( \sum_{1 \leq k \leq n, k \neq j} (\b \xi (S_k,k)-\lm (\b ))\rig)\;,
\; \; \; Z_{n,j}=P[e_{n,j}]. 
$$ 
Clearly $Q^\xi_{j}[\ln Z_{n,j}]=Q^\xi_{j-1}[\ln Z_{n,j}]$, and hence, 
\bdnn 
V_{n,j} 
 &=& Q^\xi_{j} \left[
\ln \frac{Z_n}
{ Z_{n,j} } \right]
 -
Q^\xi_{j-1} \left[
\ln \frac{Z_n}
{ Z_{n,j}} \right]\;.
\ednn 

By (\ref{gCH}) in Lemma \ref{Lem.gCH} with $X_{\cdot} =  \b \xi (\cdot,j)-
\lm (\b )$
and $\alpha_{\cdot} = 
\frac{ P[ e_{n,j} : S_j= \cdot ]}
{ Z_{n,j}}$, we see that
\bdnn
 -
Q^\xi_{j-1} \left[
\ln \frac{Z_n}
{ Z_{n,j}} \right] 
& = & 
-Q^\xi_{j-1}\lef[ Q \lef[ 
\lef.\ln \lef( \sum_x \a_x \exp (X_x) \rig) \rig| \cE_{n,j}\rig] \rig]\\
& \in & [0, c]\;,
\ednn
where $\cE_{n,j}=\s [\xi (\cdot, k)\; ; \; 1 \leq k \leq n, k \neq j]$; 
note that $\a_\cdot$ is $\cE_{n,j}$-measurable. 
Therefore, for $\tht \in \R$, we have by Jensen's inequality that 
\bdnn 
Q[ \lef.\exp \tht V_{n,j} \rig| \cE_{n,j}] 
& \leq &  
e^{c\tht^-} Q\left[ \lef.\exp  \tht Q^\xi_{j} \left[
\ln \frac{Z_n}
{ Z_{n,j}} \right] \rig| \cE_{n,j}\right]\\
& \leq &  e^{c\tht^-}Q\left[ \lef.\left( \frac{Z_n}
{ Z_{n,j}}\right)^\tht \rig| \cE_{n,j}\right] \\
& = & e^{c\tht^-}Q\left[ \lef. \left( \sum_x \alpha_x \exp (X_x)
\right)^{\tht} \rig| \cE_{n,j}\right] \\
& \leq & 
e^{c\tht^-}\sum_x \alpha_x  
Q\lef[ \lef. \exp (\tht X_x) \rig| \cE_{n,j}\rig] \\
& = & 
%%%e^{c\tht^-}\sum_x \alpha_x Q[ \exp (\tht X_x)|\cE_{n,j}] \\
e^{c\tht^-}\sum_x \alpha_x Q[ \exp (\tht X_x)]\\
& = & \exp\{c\tht^- +\lm (\tht\b )-\tht \lm (\b )\},
\ednn
where $\tht^-=\max \{ 0,-\tht\}$. Finally we conclude that 
$$
Q[ \exp |V_{n,j}| ] \leq Q[ \exp(V_{n,j}) + \exp(-V_{n,j})] \leq c_1 < \8\;.
$$
Therefore, the large deviation estimate for sum of martingale-differences 
of Lesigne and Voln\'y (2001,  Theorem 3.2)
%%%(\cite{LesVol01}, Theorem 3.2)  
applies to our case, yielding  (\ref{concent}). 
The final statement in \Prop{press} is now a simple consequence of 
Borel-Cantelli lemma. 
%%%%%%%%%%%% 
\hfill $\Box$

%%%%%%%%%%%%
\Remark{press2}
%%%%%%%%
Let us remark that the stronger assumption (\ref{exp2}) 
implies the exponential concentration (\ref{concent2}). 
In what follows, 
$c_i=c_i (\b, \del)$ ($i=1,2,\ldots $) are positive constants. 
Note first that (\ref{exp2}) implies that
$ \lm (\b ) \le c_1 +c_2\b^2 $ for all $\b \in \R$ 
and hence that
\bdnl{thtVj}
Q \left[ \exp (\tht V_{n,j})|\cE_{n,j} \right] \le \exp 
\lef( c_3 +c_4\tht^2 \rig), 
\; \; \; \mbox{for all $\tht \in \R$,}
\edn
by the computations in our proof of \Prop{press}.
By expanding the exponential and using the fact $Q[V_{n,j}|\cE_{n,j}]=0$, 
we can improve (\ref{thtVj}) into the following stronger form;
\bdnl{expaVj}
Q \left[ \exp (\tht V_{n,j})|\cE_{n,j} \right] \le \exp 
\lef( c_5\tht^2 \rig), 
\; \; \; \mbox{for all $\tht \in \R$.}
\edn
It is then, not difficult to conclude  
(\ref{concent2}) by (\ref{expaVj}) 
and a standard Gaussian estimate for a martingale.
%%%%%%%%%%%%
\end{remark}
%%%%%%%%%%%

\vs

{\bf Acknowledgements:}  We want to thank Philippe
Carmona and Yueyun Hu for 
sending us their preprint 
%%\cite{CaHu01} 
(2001)
prior its publication.
FC thanks the Graduate School of Science 
of Kyoto University for its hospitality.
FC is partially supported by CNRS, UMR 7599
``Probabilit{\'e}s et Mod{\`e}les
Al{\'e}atoires''.
TS is partially supported by JSPS Grant-in-Aid for Scientific
Research, Kiban(C) 13640103.
NY is partially supported by JSPS Grant-in-Aid for Scientific
Research, Shorei(A) 40240303.
                                  

%%%%%%%%%%%%%%
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%%%%%%%%%%%%

}

\end{document}


\newpage 

\appendix


%%%%%%%%%%%
\SSC{Appendix:List of major changes in the present version}
%%%%%%%%%%%
This appendix should eventually be removed, of course.
\bitemize
\item $Z_{n,j}$ in the proof of \Prop{press}
 \item Rem.5.1
 \item I thank Carmona-Hu for giving me their paper; I used "we ", is
   it OK~? Yes it is.
\eitemize


\end{document}


%%%%%%%%%%%%%%%%%%%%%%%%
\bibitem[Wut98]{Wut98} 
 W{\"u}thrich, Mario V. Superdiffusive behavior of two-dimensional
 Brownian 
motion in a Poissonian potential. 
Ann. Probab. {\bf 26}, no. 3, 1000--1015,  (1998).
%%%%%%%%%%%%%%
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