Content-Type: multipart/mixed; boundary="-------------0201290243243" This is a multi-part message in MIME format. ---------------0201290243243 Content-Type: text/plain; name="02-41.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-41.keywords" Directed polymers, martingales, random environment ---------------0201290243243 Content-Type: application/x-tex; name="csy" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="csy" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % 23.01.02 : version submited to Bernoulli SANS CORRESPONDING AUTHOR % % Directed Polymers in Random Environment: % Path Localization and Strong Disorder %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[12pt]{article} %%%%%%%%%%% print the keys for \ref %%%%%%%%%%%\usepackage{showkeys} \addtolength{\topmargin}{-10ex} \addtolength{\topskip}{0pt} \setlength{\oddsidemargin}{0pt} 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{\omega}} \newcommand{\om}{\omega} \newcommand{\W}{\Omega} \newcommand{\Om }{\Omega} \newcommand{\vW}{\varOmega} \newcommand{\cA }{{\cal A}} \newcommand{\cB }{{\cal B}} \newcommand{\cC }{{\cal C}} \newcommand{\cD }{{\cal D}} \newcommand{\cE }{{\cal E}} \newcommand{\cF }{{\cal F}} \newcommand{\cG }{{\cal G}} \newcommand{\cH }{{\cal H}} \newcommand{\cI }{{\cal I}} \newcommand{\cJ }{{\cal J}} \newcommand{\cK }{{\cal K}} \newcommand{\cL }{{\cal L}} \newcommand{\cM }{{\cal M}} \newcommand{\cN }{{\cal N}} \newcommand{\cO }{{\cal O}} \newcommand{\cP }{{\cal P}} \newcommand{\cQ }{{\cal Q}} \newcommand{\cR }{{\cal R}} \newcommand{\cS }{{\cal S}} \newcommand{\cU }{{\cal U}} \newcommand{\cV }{{\cal V}} \newcommand{\cW }{{\cal W}} \newcommand{\pii}{\pi \sqrt{-1}} \newcommand{\epty}{\emptyset} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %%%%%%%%%%%%%%%%%%%% suppress numbers in bibliography \makeatletter \renewcommand{\@biblabel}[1]{} \makeatother %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \makeatletter \def\section{\@startsection{section}{1}{\z@}{-3.5ex plus -1ex minus -.2ex}{2.3ex plus .2ex}{\bf}} \def\subsection{\@startsection{subsection}{2}{\z@}{-3.25ex plus -1ex minus -.2ex}{1.5ex plus .2ex}{\bf}} \makeatother % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % BEGINNING OF TEXT % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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{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{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{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{B0 \}=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{