\magnification=1200 \input amstex.tex \input amsppt.sty \vsize 17.7cm \scrollmode \baselineskip 15pt \overfullrule=0pt \def\bx{{\bf X}} \def\a{\alpha} \def\th{\theta} \def\be{\beta} \def\vep{{\varepsilon}} \def\de{{\delta}} \def\la{{\lambda}} \def\ga{{\gamma}} \def\Ga{{\Gamma}} \def\si{{\sigma}} \def\de{{\delta}} \def\ze{{\zeta}} \def\om{{\omega}} \def\Om{\Omega} \def\up{\Upsilon} \def\nul{{\nu_\la}} \def\nua{\nu_\a} \def\nur{{\nu_\rho}} \def\mur{{\mu_\rho}} \def\nurl{{\nu_{\rho,\la}}} \def\ofp{(\Omega, $ $ {\cal F}, $ $ P)} \def\lar{{\leftarrow}} \def\w{{\wedge}} \def\cX{{\cal X}} \def\cY{{\cal Y}} \def\cL{{\cal L}} \def\xil{{\it et al. }} \def \ie{{\it i.e.}} \def\limt{\lim_{t\to\infty}} \def\limn{\lim_{n\to\infty}} \def\lime{\lim_{\vep\to 0}} \def\proof{\noindent{\bf Proof. }} \def\one{{\bold 1}} \def\today{\rightline{\ifcase\month\or January\or February\or March\or April\or May\or June\or July\or August\or September\or October\or November\or December\fi \space\number\day,\space\number\year}} %%%%%%%%%%%%%%%%% EQUAZIONI CON NOMI SIMBOLICI %%% %%% per assegnare un nome simbolico ad una equazione basta %%% scrivere \Eq(...) o, in \eqalignno, \eq(...) o, %%% nelle appendici, \Eqa(...) o \eqa(...): %%% dentro le parentesi e al posto dei ... %%% si puo' scrivere qualsiasi commento; %%% per assegnare un nome simbolico ad una figura, basta scrivere %%% \geq(...); per avere i nomi %%% simbolici segnati a sinistra delle formule e delle figure si deve %%% dichiarare il documento come bozza, iniziando il testo con %%% \BOZZA. Sinonimi \Eq,\EQ,\EQS; \eq,\eqs; \Eqa,\Eqas;\eqa,\eqas. %%% All' inizio di ogni paragrafo si devono definire il %%% numero del paragrafo e della prima formula dichiarando %%% \numsec=... \numfor=... (brevetto Eckmannn); all'inizio del lavoro %%% bisogna porre \numfig=1 (il numero delle figure non contiene la sezione. %%% Si possono citare formule o figure seguenti; le corrispondenze fra nomi %%% simbolici e numeri effettivi sono memorizzate nel file \jobname.aux, che %%% viene letto all'inizio, se gia' presente. E' possibile citare anche %%% formule o figure che appaiono in altri file, purche' sia presente il %%% corrispondente file .aux; basta includere all'inizio l'istruzione %%% \include{nomefile} %%% %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \global\newcount\numsec\global\newcount\numfor \global\newcount\numfig \gdef\profonditastruttura{\dp\strutbox} \def\senondefinito#1{\expandafter\ifx\csname#1\endcsname\relax} \def\SIA #1,#2,#3 {\senondefinito{#1#2} \expandafter\xdef\csname #1#2\endcsname{#3}\else \write16{???? ma #1,#2 e' gia' stato definito !!!!} \fi} \def\etichetta(#1){(\veroparagrafo.\veraformula) \SIA e,#1,(\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\letichetta(#1){\veroparagrafo.\veraformula \SIA e,#1,{\veroparagrafo.\veraformula} \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ Sta \equ(#1) == #1}} \def\tetichetta(#1){\veroparagrafo.\veraformula %%%%copy four lines \SIA e,#1,{(\veroparagrafo.\veraformula)} \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ tag \equ(#1) == #1}} \def \FU(#1)#2{\SIA fu,#1,#2 } \def\etichettaa(#1){(A\veroparagrafo.\veraformula) \SIA e,#1,(A.\veroparagrafo.\veraformula) \global\advance\numfor by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ EQ \equ(#1) == #1 }} \def\getichetta(#1){Fig. \verafigura \SIA e,#1,{\verafigura} \global\advance\numfig by 1 % \write15{\string\FU (#1){\equ(#1)}} \write16{ Fig. \equ(#1) ha simbolo #1 }} \newdimen\gwidth \def\BOZZA{ \def\alato(##1){ {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\hsize\kern-1.2truecm{$\scriptstyle##1$}}}}} \def\galato(##1){ \gwidth=\hsize \divide\gwidth by 2 {\vtop to \profonditastruttura{\baselineskip \profonditastruttura\vss \rlap{\kern-\gwidth\kern-1.2truecm{$\scriptstyle##1$}}}}} } \def\alato(#1){} \def\galato(#1){} \def\veroparagrafo{\number\numsec}\def\veraformula{\number\numfor} \def\verafigura{\number\numfig} \def\geq(#1){\getichetta(#1)\galato(#1)} \def\Eq(#1){\eqno{\etichetta(#1)\alato(#1)}} \def\teq(#1){\tag{\tetichetta(#1)\hskip-1.6truemm\alato(#1)}} %%%%%this line \def\eq(#1){\etichetta(#1)\alato(#1)} \def\Eqa(#1){\eqno{\etichettaa(#1)\alato(#1)}} \def\eqa(#1){\etichettaa(#1)\alato(#1)} \def\eqv(#1){\senondefinito{fu#1}$\clubsuit$#1\else\csname fu#1\endcsname\fi} \def\equ(#1){\senondefinito{e#1}\eqv(#1)\else\csname e#1\endcsname\fi} %next six lines by paf (no responsibilities taken) \def\Lemma(#1){Lemma \letichetta(#1)\hskip-1.6truemm} \def\Theorem(#1){{Theorem \letichetta(#1)}\hskip-1.6truemm} \def\Proposition(#1){{Proposition \letichetta(#1)}\hskip-1.6truemm} \def\Corollary(#1){{Corollary \letichetta(#1)}\hskip-1.6truemm.} \def\Remark(#1){{\noindent{\bf Remark \letichetta(#1)\hskip-1.6truemm.}}} \let\ppclaim=\plainproclaim \let\EQS=\Eq\let\EQ=\Eq \let\eqs=\eq \let\Eqas=\Eqa \let\eqas=\eqa %%%%%%%%%%%%%%%%%% Numerazione verso il futuro ed eventuali paragrafi %%%%%%% precedenti non inseriti nel file da compilare \def\include#1{ \openin13=#1.aux \ifeof13 \relax \else \input #1.aux \closein13 \fi} \openin14=\jobname.aux \ifeof14 \relax \else \input \jobname.aux \closein14 \fi %\openout15=\jobname.aux %%%%%%%%%%%%%%%%%%%%%%%%%%%% %\BOZZA % macros extras %\let\@=@ %this is necessary because in mssymb @ is an undefined control sequence %\input mssymb \def\E{{\Bbb E}} \def\P{{\Bbb P}} \def\Z{{\Bbb Z}} \def\V{{\Bbb V}} \def\M{{\cal M}} \def\({\left(} \def\){\right)} \def\square{\ifmmode\sqr\else{$\sqr$}\fi} \def\sqr{\vcenter{ \hrule height.1mm \hbox{\vrule width.1mm height2.2mm\kern2.18mm\vrule width.1mm} \hrule height.1mm}} % This is a slimmer sqr. \rightline {\today} \centerline {\bf Asymmetric conservative processes with random rates} \vskip 2truemm \centerline { I. Benjamini, P.\ A.\ Ferrari, C. Landim} \vskip 1truemm \centerline {\it Cornell University, Universidade de S\~ao Paulo and IMPA} \vskip 1truemm \noindent {\bf Summary.} We study a one dimensional nearest neighbor simple exclusion process for which the rates of jump are chosen randomly at time zero and fixed for the rest of the evolution. The $i$-th particle's right and left jump rates are denoted $p_i$ and $q_i$ respectively; $p_i+q_i=1$. We fix $c\in (1/2,1)$ and assume that $p_i\in[c,1]$ is a stationary ergodic process. We show that there exists a critical density $\rho^*$ depending only on the distribution of $\{p_i\}$ such that for almost all choices of the rates and a (fixed) density $\rho^*< \rho \le 1$ there exists an invariant distribution for the process as seen from a tagged particle with asymptotic density $\rho$. Under this measure, the distribution of the distances between particles are independent random variables. We also show that under the invariant distribution, the position $X_t$ of the tagged particle at time $t$ can be sharply approximated by a Poisson process. Finally, we prove the hydrodynamical limit for zero range processes with random rate jumps. \vskip 5truemm \noindent {\it Keywords and phrases.} Asymmetric simple exclusion. Random rates. Law of large numbers. Hydrodynamical limit. \vskip 2truemm \noindent {\it AMS 1991 Classification.} 60K35, 82C22, 82C24, 82C41. \vskip 2truemm \noindent {\it Short title:} Asymmetric processes with random rates. \vskip 3truemm \noindent {\bf 1. Introduction.} \vskip 2truemm \numsec=1\numfor=1 Let $c\in (1/2,1)$ and $p=\{p_i\in [c,1]: i\in\Bbb Z\}$ be a stationary ergodic process on $[c,1]^\Bbb Z$. Let $m$ be their joint distribution. For each random choice of $p$ we consider an asymmetric simple exclusion process whose particles move with rates $p$. In fact we consider the simple exclusion process as seen from a tagged particle. The tagged particle is labeled zero and the other particles labeled in an increasing way according to their position. In words, the $i$-th particle attempts to jump to the nearest site to the right at rate $p_i$ and performs the jump if the destination site is empty, otherwise it stays still. At rate $q_i=1-p_i$, the $i$-th particle does the same to the left. Let $x_i(t)$ be the position of the $i$-th particle at time $t$. Let $\eta_t$ be the set of occupied sites at time $t$; hence $\eta_t = \{x_i(t): i\in\Bbb Z\}$. The configuration $\eta$ can be interpreted as a function from $\Bbb Z$ in $\{0,1\}$ given by $\eta(x) = 1\{x=x_i,$ for some $i\}$. The generator of the process seen from the tagged particle depends on the choice of $p$ and is given by $$ \eqalign { L_pf(\eta) &= \sum_{i\ne 0} \big( 1\{x_{i+1}>x_i+1\} p_i [f(\eta^{x_i,x_i+1}) - f(\eta)] \cr &\quad +1\{x_{i-1}1\} p_0 [f(\tau_{1}\eta^{0,1}) - f(\eta)] \cr &\quad +1\{x_{-1}<-1\} q_0 [f(\tau_{-1}\eta^{0,-1}) - f(\eta)] \cr} \Eq(L) $$ where $\tau_x$ is the translation by $x$: $\tau_x\eta(z)= \eta(z-x)$, $f$ is a local function and $$ \eta^{x,y}(z) = \cases \eta(z) &\text{if $z\ne x,y$}\cr \eta(x) & \text{if $z=y$}\cr \eta(y) & \text{if $z=x$}\cr \endcases $$ is the configuration where the values at the sites $x$ and $y$ have been interchanged. The existence of this process follows either from the Hille-Yosida Theorem as in Liggett (1985) and Andjel (1982) or using the graphical construction of Harris. See Ferrari (1992) for instance. For a measure $\mu$ on $\{0,1\}^\Bbb Z$ let $\lim_{n\to\infty} (2n+1)^{-1}\int d\mu(\eta) \sum_{x=-n}^n \eta(x)$ be its global density (if the limit exists). In Section 2 we show that there exists a critical density $\rho^*$ such that for any density $\rho\in (\rho^*,1]$ and almost all choices of the rates $p$, there exists an invariant measure with global density $\rho$. $\rho^*$ might be strictly positive as is shown by an example and there are no invariant product measure for densities in $[0,\rho^*)$. We then show that for any given $\rho$ and for almost all choice of the rates, the position $X_t$ of the particle originally at the origin is given by a Poisson process of rate $v(\rho,p)$ plus an error of order one. The ``effective velocity'' $v(\rho,p)$ is given by an explicit equation. In the simple exclusion process as seen from a tagged particle with $p_i \equiv p_0$, the only extremal invariant measure with global density zero is the measure concentrating mass in the configuration with a single particle at the origin. On the other hand, if one starts the system with an ergodic measure such that the inter particle distances have a translation invariant distribution and the mean distance between two successive particles is infinite, then the process converges weakly to the configuration with a single particle at the origin (Andjel, Cocozza-Thivent and Roussignol (1986)). Based in this result we conjecture that in our case there are no extremal invariant measures with global density zero concentrating mass in the set of configurations with infinitely many particles. On the other hand, we show that there are measures concentrating mass in sets of configurations with $n$ particles, for all $n$. We also study the process with initial configuration $\dots111000\dots$. In this case assuming that $m([c,c+\vep])>0$ for all $\vep>0$, we show that the $i$-th particle to the left has an asymptotic velocity that converges as $i\to\infty$ to $2c-1$. When the rates are associated to the sites instead of the particles our result is weaker: we show that starting with the homogeneous product distribution $\nur$, for almost all choices of the environment any tagged particle has a velocity at most $(2c-1)(1-\rho)$ almost surely. In the last two sections we turn to the study of the hydrodynamical behavior of the process with random rates in infinite volume. We first change coordinates and denote by $\xi_t(k)$ the number of holes at time $t$ between particle $k$ and particle $k+1$. This transform the asymmetric simple exclusion process to a so--called zero range process. It is easy to check that $\xi_t$ evolves as follows. At rate ${\bold 1}\{ \xi(k)\ge 1\} q_{k+1}$ one particle jumps from site $k$ to site $k+1$ and at rate ${\bold 1}\{ \xi(k)\ge 1\} p_{k}$ one particle jumps from site $k$ to site $k-1$. Under some restrictions on the initial measure, we prove that $m$ almost surely the macroscopic behavior of this process is described by the entropy solution of a first order hyperbolic equation $ \partial_t \rho + \partial_x v(\rho)=0$ where $v(\rho)$ denotes the mean velocity of a particle for a process in equilibrium with density $\rho$. In the last section we consider the hydrodynamical behavior of another type of zero range processes in random environment. This system corresponds to asymmetric simple exclusion processes where the rate at which the $k$-th particle jumps to the right and the rate at which the $k+1$-th particle jumps to the left is decelerated by a factor $p_k$ picked from an ergodic stationary process. \vskip 3truemm \noindent {\bf 2. Asymmetric exclusion with random rates associated to particles.} \vskip 2truemm \numsec=2\numfor=1 Without loss of generality we shall asume throughout this section that $c$ is such that $$ m([c,c+\epsilon))\; >\; 0 \quad\text{for all}\quad \epsilon \; >\; 0\; .\Eq(c+ep) $$ For an integer $i$, denote by $Z_i$ the (eventually infinite) function of $p$: $$ Z_i(p)\; =\; { 1\over q_{i+1}} \sum_{j=0}^\infty \prod_{k=i-j}^i {q_{k+1}\over p_{k}}\; . $$ Let $v^*$ the maximum value of $v$ for which $Z_i<(1/v)$ almost surely for all $i\in\Bbb Z$~: $$ %[v^*]^{-1} \; =\; \inf\{v>0: m(Z_i \ge v)=0\} v^* \; =\; \sup\{v>0: m(v Z_i < 1)=1\} %v^* \; =\; \inf\{v>0: m(Z_i \ge v)=0\} %v^*\; =\; \sup \Big\{ v>0;\; Z_i \; <\, 1/v\quad %\forall i\in\Bbb Z \quad a. s.\Big\}\; . $$ Notice that $v^*\ge 2c-1$. Define the function $R:[0,v^*)\to\Bbb R_+$, by $$ R(w) = \left(m \left[ \left(1- w Z_i \right)^{-1}\right]\right)^{-1} \Eq(rvc) $$ that does not depend on $i$ as $p$ is a stationary process. Notice that by the definition of $v^*$, the function $R$ is well defined and positive in $[0,v^*)$. Moreover, $R(0)=1$ and $R$ is clearly decreasing in $w$. In particular it is invertible. Let $v$ denote the inverse function: $v(R(w))=w$. Denote by $R(v^*)\ge 0$ the limit of $R(w)$ as $w\uparrow v^*$. \ppclaim \Theorem (im). For any $\rho\in [R(v^*),1]$ and for almost all choice of $p$: (i) There exists a unique vector $\a (= \a(p,\rho))= \{\a_i: i\in\Bbb Z\}$, such that for all $i$, $$ \eqalign{ &\a_i p_i - \a_{i-1} q_i = v(\rho)\cr & \a_i\in [0,1).\cr} \Eq (al) $$ (ii) The measure $\nua$ defined by $$ \nua( x_{i+1} - x_i = k_i: i \in A) = \prod_{i \in A} \a_i^{k_i-1} (1-\a_i), \ \ A\subset \Bbb Z,\ \ k_i\ge 1, \text{ finite} \Eq(nua) $$ with $\nu_\a ( x_0=0)=1$ is invariant for the process with generator $L_p$. (iii) The global density of this measure is $\rho$: $$ \limn {1\over 2n+1}\sum_{x=-n}^n \eta(x) = \rho \Eq(gd) $$ $\nua$ almost surely. \hfill\break (iv) If $\rho < R(v^*)$, there are no invariant measures for $L_p$ with global density $\rho$ for which $\{x_i - x_{i-1}\}_i$ are independent. \noindent {\bf Remarks.} If the vector $p$ is constant, that is $p_i\equiv c$, then $v(\rho) = (2c-1)(1-\rho)$ and $\rho^*=0$. On the other hand, this theorem implies that there are product invariant measures with global densities $\rho$ {\it only} for $\rho>\rho^* = R(v^*)$. The critical density $\rho^*$ may be strictly positive as is shown by an example below. It is not discarded that it could be invariant measures that are not product with global densities below $\rho^*$. It is an open problem to describe in some sense the evolution of the process starting from the Bernouilli product measure with density below the critical density. \medskip \noindent {\bf Proof.} Fix $0\le v<2c-1$. We consider the (non homogeneous) dynamical system $a_i$ with state space $[0,1]$ defined by $$ a_{i} = {v+ a_{i-1} (1-p_i) \over p_i} := T_i(a_{i-1})\; . $$ Of course $T_i$ depends on $p_i$. In order to argue that the dynamical system $a_i$, $i\in\Bbb Z$ has a unique solution satisfying \equ(al) we consider a coupling $(a_i,b_i)$ between two realizations of the process starting from some $j$ with different initial conditions $a_j$ and $b_j$. To realize the coupling we use the same $p_i$ for both processes and obtain for $i\ge j$, $$ b_i - a_i = { (b_{i-1} - a_{i-1}) (1-p_i) \over p_i} \le (b_{j} - a_{j})\left(\shave{1- c \over c}\right)^{i-j} \Eq(a-b) $$ From \equ(a-b) we learn two things. First, that the process is attractive: $b\ge a$ implies $T_ib\ge T_ia$ for any $i$. This implies that if we consider $a_j = 0$ and $b_j = 1$, then for any $c_0 \in [0,1]$, it holds $a_i \le c_i \le b_i$ for $i\ge 0$. Second, for any initial condition $a_j1/2$. On the other hand, \equ(a-b) implies that if we fix the sequence $p$ and call $\a_i(n)$ the $(n+i)$-th iterate of the chain when $\a_{-n} = 1$: $$ \a_i(n) = T_{i-1}T_{i-2}\dots T_{-n} (1) $$ for $i>-n$. Then $\a_i(n)$ is a non decreasing sequence in $n$ and it converges exponentially fast, as $n\to\infty$, to a limit $\a_i$. We have proved that the double infinite sequence $\a$ is the unique solution of \equ(al). This unique solution is given by $$ \a_i(p,v) = \a_i(p) = {v \over q_{i+1}} \sum_{j=0}^\infty \prod_{k=i-j}^i {q_{k+1}\over p_{k}}\; . \Eq(alpha) $$ Recall the definition of $v^*$. From this explicit formula for $\a_i$, we see that we may extend the domain of $v$ for which there exist a solution of \equ(al) to the set $[0,v^*)$. Stationarity of $\{p_i\}$ implies that the process $\{\a_i\}$ is stationary. The easiest way to show (ii) is to show $$ \nua (gL_p f) = \nua (fL^*_p g) $$ for $f,g$ in a core of $L$, where the (guessed) adjoint generator $L^*_p$ is defined by $L_p^* = L_{p^*}$, with $$ p^*_i = { q_i \a_{i-1} \over \a_i},\ \ q^*_i = { p_i \a_i \over \a_{i-1}} $$ It is sufficient to take $f$ and $g$ of the form $1\{x_{i+1} - x_i = k_i: i \in A\}$, with $k_i\ge 1$ and $A$ a finite subset of $\Bbb Z$. Since the measure is product when applied to this kind of functions, the problem is reduced to show that for all $i$ and all $k_i, k_{i-1}\ge 0$, $$ p_i\nua (x_{i+1}- x_i = k_i, x_i - x_{i-1} = k_{i-1}+1)= q^*_i\nua (x_{i+1}- x_i = k_i+1, x_i - x_{i-1} = k_{i-1}) $$ and $$ p^*_i + q^*_{i+1} = 1 $$ that immediately hold using \equ(al). The invariance of $\nua$ has been proven by Jackson (1963) for finite systems and by Andjel (1982) for infinite systems. The proof using the adjoint can be found in Kipnis (1986) for the exclusion process with constant rates and (for a more complicate model) in Ferrari and Fontes (1994a). To show (iii) we first look for the solution of \equ(al). For a fixed realization $p$ the (unique) solution of the system \equ(al) is given by \equ(alpha). Since $p$ is ergodic, so must be $\{a_i\}$ and calling $mh(a_i) = \int m(dp) h(a_i(p))$, $$ \limn {1\over 2n+1}\sum_{k=-n}^n {1\over 1- a_k} = m(1/(1-a_i)),\ \ \ m \text{ almost surely}. \ \Eq(bar) $$ that by stationarity does not depend on $i$. Call $m\nua$ the measure on $\{0,1\}^\Bbb Z$ obtained by first choosing a $p$ according to $m$ and then for $i\in\Bbb Z$ choosing $x_{i+1}-x_i$ with geometric distributions as in \equ(nua) with $\a=\a(p)$. It is immediate that $m\nua((x_{i+1}-x_i)^2)-(m\nua(x_{i+1}-x_i))^2<$ constant and $m\nua(x_{i+1}-x_i) = m(1/(1-\a_i))$. Hence, $$ \limn {1\over 2n+1}\sum_{i=-n}^n (x_{i+1}-x_i - m (1/(1-\a_i)) = 0 \ \ m\nua,\text{ almost surely}. $$ (See exercice 11 of page 314 of Grimmett and Stirzaker (1992) for instance.) which in turn implies again by ergodicity, $$ \limn {1\over 2n+1}\sum_{x=-n}^n \eta(x) = {1 \over m(1/(1-\a_i))}.\ \ m\nua, \text{ almost surely} $$ To show (iv) we first observe that invariant measures for which $\{x_i - x_{i-1}\}$ are independent must be product of geometric as in \equ(nua). This follows from direct computation with the generator. On the other hand, if the measure concentrates in configurations with infinitely many particles, the probability $\a_i$ that particles $i$ and $i+1$ are at distance bigger than one must satisfy equation \equ(al) for some $v\in [0,v^*)$. The global density of this measure is $R(v)$ that is necessarily bigger than $\rho^*$. \square \medskip We proved in Theorem \equ(im) that there exist product invariant measures for all densities $\rho$ greater than a critical density $\rho^*= R (v^*)$. We present below an example showing that the critical density $\rho^*$ may be strictly positive. \medskip \noindent {\bf Example. The independent case.} Assume that $p_i$ are i.i.d. random variables taking only two values $c$ and $1$~: $m[p_0=c]=\theta = 1-m[p_0=1]$. For each integer $i$, denote by $T_i$ the distance from $i$ to the first integer $j$ on the left of $i$ with $p_j=1$~: $$ T_i\; =\; \min\Big \{ j\ge 0;\; p_{i-j}=1\Big\}\; . $$ From formula \equ(al) it is easy to see that $\a_i$ may be expressed as a simple function of $T_i$~: $$ \a_i\; =\; \frac{v}{2c-1} \Big\{ 1- 2(1-c) \left(\frac{1-c} {c}\right)^{T_i}\Big\}\; . $$ Since $c>1/2$ and $T_i$ may assume with positive probability large values, $v^*=2c-1$. Furthermore, since $T_i$ has a geometric distribution (\ie, $m(T_i \ge k) = \theta^k$, $k \ge 0$), $[R(v)]^{-1}$ can be computed explicitely: $$ \frac{1}{R(v)}= m\left( {1\over 1-\a_i}\right) = \sum_{k\ge 0} { (1-\theta) \theta^k \left[ 1 - {v \over 2c-1} \left(1-2(1-c)\left({1-c \over c}\right)^k\right)\right]^{-1}} \; . $$ This is positive for all $v \le 2c -1$. In the limit as $v \to 2c-1$ we get $$ \lim_{v\to 2c-1} [R(v)]^{-1} =\frac{1-\theta}{2(1-c)} \sum_{k\ge 0} \left[ \frac{\theta c}{1-c}\right]^k \; <\; \infty $$ provided $\theta <(1-c)/c$. Hence, if $\theta < {1-c \over c}$ we can compute the sum and obtain $$ \rho^* = {2(1-c(1+\theta)) \over 1-\theta} $$ In this example therefore there are invariant product measures with global density $\rho$, only for $\rho\ge R(2c-1)=\displaystyle{ {2(1-c(1+\theta)) \over 1-\theta}>0}$ for $\theta< {1-c \over c}$. Figures 1 and 2 ilustrate this point. In particular we can think that if we perturb an homogeneous system with constant rates $p_i \equiv 1$, in such a way that a small fraction $\theta$ of the $p_i$'s are modified to be $c$, then this perturbed system will not have product invariant measures with small global densities. \input pictex $$ \beginpicture \setcoordinatesystem units <.5in,.5in> \setplotarea x from -5.0 to 3.0 , y from -1 to 2.5 \setplotsymbol({\sevenrm .}) %left picture \arrow <6pt> [.2, .7] from -5.3 0 to -2 0 \arrow <6pt> [.2, .7] from -5 -0.3 to -5 2.5 \put{{\it Figure 1:} $\theta\ge {1-c \over c}$} at -4 -.7 \put{$0$} at -5.15 -.15 \put{$2c-1$} at -3 -.20 \put{$R(v)$} at -5.4 2 \put{$1$} at -5.15 1.6 \put{$v$} at -2.2 -.2 %\setplotsymbol({\sevenrm .}) \setquadratic \plot -5 1.6 -4 .5 -3 0 / \plot -3 .05 -3 0 -3 -.05 / %right picture \arrow <6pt> [.2, .7] from -0.3 0 to 3 0 \arrow <6pt> [.2, .7] from 0 -0.3 to 0 2.5 \put{{\it Figure 2:} $\theta< {1-c \over c}$} at 1 -.7 \put{$0$} at -.15 -.15 \put{$2c-1$} at 2 -.20 \put{$\rho^*$} at -.18 .4 \plot -.05 .4 0 .4 .05 .4 / \put{$R(v)$} at -.4 2 \put{$1$} at -.15 1.6 \put{$v$} at 2.8 -.2 %\setplotsymbol({\sevenrm .}) %\setquadratic \plot 0 1.6 1 .7 2 0.4 / \plot 2 .05 2 0 2 -.05 / \endpicture $$ \ppclaim \Theorem (x=p). Let $\rho\ge\rho^*$ and take $p$ in the set of $m$--probability one for which there exists $\nua$, the invariant measure defined in Theorem \equ(im) with global density $\rho$ and $\a=\a(p)$. Assume that the initial distribution of particles is given by $\nua$ and let $X_t$ be the position at time $t$ of the tagged particle originally at the origin. Then there exist Poisson processes $N^0_t$ and $N^1_t$ of rate $v=v(\rho)$, random variables $R^0$ and $R^1$ and a stationary process $B_t$ such that the following inequalities hold $$ N^0_t + R^0 \ge X_t \ge N^1_t +B^1_t - B^1_0 - R^1 $$ Furthermore $B_t$, $R^0$ and $R^1$ can be chosen such that they have an exponential moment independent of $p$ (and $t$): there exist positive $\theta$ and $C$ independent of $p$ and $t$ such that $$ \E e^{\theta R^0}< C,\ \E e^{\theta R^1}< C, \ \E e^{\theta B_t} 0) - (1-b)\mu^n (z_{j+1}>0) = {2b-1 \over 1- \left(\displaystyle {1-b \over b}\right)^{n+1}} \Eq(bn) $$ that does not depend on $j$. Furthermore we have that if the system starts with the configuration $z_j \equiv 0$ (that has positive $\mu^n$ measure), it holds the following law of large numbers $$ \limt {y_j(t) - y_j(0)\over t} = v^n. $$ Turning to the semi-infinite system, fix $p$. It is clear that $v_i \le v_{i+1}$. On the other hand, let $b(i,n,p) = \max \{p_j: i\le j \le i+n\}$. For this $b$ consider the finite system $(y_0,\dots,y_n)$ defined above with initial positions $(x_i,\dots,x_{i+n})$. Hence one can couple in such a way that $$ x_i(t) - x_i(0) \le y_0(t) - y_0(0) $$ and $$ \limsup_{t\to\infty} {x_i(t) - x_i(0)\over t} \le \limt {y_0(t) - y_0(0)\over t} = {2b(i,n,p) -1\over 1- \left({ 1-b(i,n,p)\over b(i,n,p)}\right)^{n+1} }, $$ by \equ(bn) and a law of large numbers for a positive recurrent Markov chain. Take $\vep >0$. Take $b$ and $n$ such that $v^n< 2c-1 + \vep$, where $v^n$ is given by \equ(bn). Let $I(p) = \max \{i: b(i,n,p) < b\}$ the first time that $n$ successive $p_j$ are smaller than $b$. By assumption \equ(c+ep) $I$ is finite with $m$-probability one. Hence we have that for all $\vep>0$ for $i$ sufficiently large, $$ 2b(i,n,p) - 1 \le 2c-1 +\vep $$ This proves the Theorem. \square When the rates are attached to the sites instead of the particles, there is no description of the invariant measure available. Presumably in this case the invariant measure is not product and it might have long correlations. A simple coupling argument shows that if $m$ concentrates in $(c,1)$ and the particles are distributed initially according to the translation invariant product measure $\nur$, then the tagged particle must move at velocity at least $(2c-1)(1-\rho)$. This is the contents of the next theorem. \ppclaim\Theorem(sites). Assume that the rates $p$ are attached at the sites and that $m$ concentrates in $[c,1]$. Let $\eta_t$ be the simple exclusion process with these rates distributed initially with the (non invariant) distribution $\nur$. Let $X_t$ be the position of the tagged particle initially located at the origin. Then $$ \liminf_{t\to\infty} {X_t \over t} \ge (2c-1)(1-\rho) \ \ \text{almost surely.} $$ \proof We couple $\eta_t$ with a simple exclusion process $\ze_t$ with (constant) rates $c$ and $1-c$ for right and left jumps respectively. The product measure $\nur$ conditioned to have a particle at the origin is invariant for this process as seen from the tagged particle. Let $Z_t$ be the position of the tagged particle initially at the origin. Under initial distribution $\nur$, Kipnis (1986) proved that $$ \limt {Z_t\over t} = (2c-1)(1-\rho). \Eq(zt) $$ Since $p_x \ge c$ for all $x$ we can couple $\eta_t$ and $\ze_t$ in such a way that each time a jump from $x$ to $x+1$ is attempted for the system $\ze_t$, the same jump is attempted for the system $\eta_t$. Analogously, since $1-c \ge q_i$, each time a jump from $x$ to $x-1$ is attempted for the system $\eta_t$, the same jump is attempted for the system $\ze_t$. Since the rates are different, under this coupling the two systems will differ. Let $x_i(t)$ and $z_i(t)$ the positions of the particles of the $\eta_t$ and $\ze_t$ systems respectively at time $t$. Here $x_0(t) = X_t$ and $z_0(t) = Z_t$. At time zero $x_i(0) \equiv z_i(0)$. We claim that for all $t\ge 0$ $$ x_i(t) \ge z_i(t) \Eq(xgz) $$ To show \equ(xgz) it suffices to see that each jump involving positions $x_i(t)$ and $z_i(t)$ keep the inequality unaltered. Assume that at time $t$ \equ(xgz) holds and that there is an event involving $x_i(t)$ and/or $z_i(t)$ in the time interval $(t,t+dt)$. If $x_i(t) > z_i(t)$ then, after the event, $x_i(t) \ge z_i(t)$ because at most a jump of length $1$ occurred. If $x_i(t) = z_i(t)$, there are four possibilities: (a) an attempt for both particles to jump to the right; (b) an attempt for the $x_i(t)$ particle to jump to the right; (c) an attempt for both particles to jump to the left and (d) an attempt for the $z_i(t)$ particle to jump to the left. Since by \equ(xgz), $x_{i+1}(t) \ge z_{i+1}(t)$, in case (a) after the jump $x_i(t+dt) \ge z_i(t+dt)$. Case (c) is similar and cases (b) and (d) are immediate because the jumps are in the good sense. This proves \equ(xgz) for all $t$. In particular $X_t \ge Z_t$. This and \equ(zt) show the theorem.\ \ \square \bigskip \def\a{\alpha} \def\b{\beta} \def\d{{\delta}} \def\e{\eta} \def\ep{\epsilon} \def\vep{{\varepsilon}} \def\vf{\varphi} \def\g{{\gamma}} \def\G{{\Gamma}} \def\l{\lambda} \def\L{{\Lambda}} \def\r{\rho} \def\s{{\sigma}} \def\th{\theta} \def\z{{\zeta}} \def\o{{\omega}} \def\O{\Omega} \def\N{{\Bbb N}} \def\R{{\Bbb R}} \def\Z{{\Bbb Z}} \noindent{ \bf 3. Hydrodynamical limit of one dimensional asymmetric zero range processes with random jumps.} \medskip \numsec=3 \numfor=1 In this section we study the hydrodynamical behaviour of the asymmetric process with random rates introduced in last section. In order to do it, we first change coordinates. For an integer $k$, define $\xi_t (k)$ to be the number of holes between the $(k+1)$-th and the $k$-th particles at time $t$ : $$ \xi_t (k)\; =\; x_{k+1}(t) \; -\; x_k(t)\; -\; 1\; . $$ It is then easy to check that for each realization $\{p_k; \; k\in\Bbb Z\}$ $\xi_t$ is a Markov process on $\N^\Bbb Z$ whose generator acts on cylinder function as $$ (L_p' f) (\xi) \; =\; \sum_{\scriptstyle j,k\in \Bbb Z\atop \scriptstyle |k-j|=1} p (k,j) {\bold 1}\{\xi(k) \ge 1\} [ f(T_{k,j}\xi) - f(\xi)] \; . \Eq (gen3) $$ In this formula, for fixed integers $j$ and $k$ and for configurations $\xi$ with at least one particle at $k$, $T_{k,j}\xi$ stands for the configuration obtained from $\xi$ letting one particle jump from site $k$ to site $j$ : $$ \Big( T_{k,j} \xi \Big ) (i) = \cases \xi(i) & \text{if $i\ne k,j$} \cr \xi(k) -1 & \text{ if $ i=k$} \cr \xi(j) + 1 & \text{ if $ i=j$.}\cr \endcases $$ Moreover, with notation introduced in last section, $p(k,k+1)= q_{k+1}$ and $p(k,k-1)=p_k$ so that $p(k,k+1) + p(k+1,k)=1$. This is the so called zero range process with jump rate $g(n)={\bold 1}\{ n\ge 1\}$ and random transition probabilities $p(j,k)$. We already obtained in last section the invariant measures of this process. Translated to our context they write as follows. For each $v$ in $[0,v^*)$, consider $\a_i(p)=\a_i(p,v)$ the solution of \equ(al) and denote by $\mu_{p,v}$ the product measure on $\N^{\Bbb Z}$ with marginals given by $$ \mu_{p,v} \{\xi ;\xi (i) = k\} \; =\; (\a_i)^k (1-\a_i) \qquad\text{for } k\in\N \text{ and } i\in \Bbb Z\; . $$ Define $M\colon [0, v^*)\to\R_+$ as the expected value of particles at site $0$ for the measure $\mu_{p,v}$ : $$ M(v)\; =\; \mu_{p,v}\big[ \eta (0)\big] \; =\; \frac{\a_0}{1-\a_0}\; . $$ We have thus that $M$ is a smooth and strictly increasing positive function with $M(0)=0$. Define by $\r \colon [0, v^*)\to\R_+$ the expected value of $M(v)$~: $$ \r (v)\; =\; m\Big[ M(v)\Big]\; . \Eq(rho) $$ From its definition and from the properties of $M$, it follows that $\r$ vanishes at $0$ and that $\r$ is continuous and strictly increasing. Moreover, with the very same arguments presented in the proof of Theorem 2.2(iii), one obtains that $m$ almost surely, $$ \lim_{n\to\infty} |\L_n|^{-1}\sum_{k\in\L_n} \xi(k) \; =\; \r (v)\qquad \mu_{p,v} \quad\hbox {almost surely}\; . $$ Here, for a positive integer $n$, $\L_n$ represents a box of length $2n+1$ centered at the origin~: $$ \L_n\; =\; \{-n,\dots ,n\} $$ and $|\L_n|$ its volume. We now introduce the initial measures considered in this section. For a bounded continuous function $\l_0\colon \R\to\R_+$ denote by $\{\mu^N_{\l_0} ; \; N\ge 1\}$ the sequence of product measures on $\N^{\Bbb Z}$ with marginals given by $$ \mu^N_{\l_0}\{\xi ; \xi (k) = n\} \;=\; \Big[ 1+\l_0 (k/N) \Big]^{-1} \left( \frac{\l_0 (k/N)} {1+\l_0 (k/N)}\right)^n $$ for $k\in\Bbb Z$ and $n\ge 0$. $\{\mu^N_{\l_0} ; \; N\ge 1\}$ is defined so that expected number of particles at $k$ is equal to $\l_0(k/N)$. To ensure that, for $m$ almost all environments $p$, $\mu^N$ is bounded above (for the natural partial order on the space of all probability measures of $\N^{\Bbb Z}$) by some invariant measure $\mu_{p,v}$, we will have to assume that the initial profile $\l_0$ is bounded above by $\sup_{v \delta \right ] \; =\; 0 $$ $m$ almost surely. In this last formula $\l$ is the unique weak entropy solution of the first order quasilinear hyperbolic equation $$ \left\{ \eqalign{ & \partial_t \l \; +\; \partial_{x} a(\l)\; =\; 0\; \cr & \l(0,\cdot)\; =\; \l_0 (\cdot)\cr} \right. $$ and $a$ is the inverse of the function $\rho$ defined in \equ(rho). \endproclaim \medskip The proof of this theorem is similar to the one of Theorem 4.2 presented in next section and thus omitted. \bigskip \subheading{ 4. Asymmetric zero range processes with random rates} \medskip \numsec=4 \numfor=1 In section 3 we studied the hydrodynamical behaviour of one dimensional zero range processes with random transition probabilities. In this section, instead, we shall study zero range processes with random rate jumps and fixed deterministic transition probabilities. As in section 1 consider a family of random variables $\{ p_k;\; k\in \Bbb Z^d\}$ with values in an interval $[c,1]$ for some strictly positive constant $c$. Denote by $m$ the joint distribution and assume that is is stationary and ergodic. We will call $p$ the environment and shall assume that $m([c,c+\epsilon))>0$ for all $\epsilon>0$. For each realization of $p$, let $\eta_t$ be the Markov process on $\N^{\Bbb Z^d}$ whose generator acts on cylinder functions as $$ (L_p^2 f) (\eta) = \sum_{k,j\in \Bbb Z^d} r(j) p_k g(\eta(k)) [ f(T_{k,k+j}\eta) - f(\eta)] \; , \Eq(3.2) $$ where $r(\cdot)$ is a finite range transition probability on $\Bbb Z^d$ and $g$ a positive non decreasing bounded function vanishing at $0$ : there exists $\Gamma<\infty$ such that $r(j)=0$ for $|j|>\Gamma$, $0=g(0) \delta \right ] \; =\; 0 $$ $m$ almost surely. In this last formula $\l$ is the unique weak entropy solution of the first order quasilinear hyperbolic equation $$ \left\{ \eqalign{ & \partial_t \l \; +\; \sum_{i=1}^d \gamma_i \partial_{x_i} a(\l)\; =\; 0\; \cr & \l(0,\cdot)\; =\; \l_0 (\cdot)\; .\cr} \right. $$ \endproclaim \medskip The proof of this theorem is analogous to the one of Theorem 1 presented in sections 5 and 6 of Landim (1993b). We therefore concentrated only on the main differences. Also, to keep notation simple, we present the proof in dimension 1 and assume that the environment takes only two values $c=c_1}\cr \vskip 3truemm \+Instituto de Matem\'atica e Estat\'\i stica --- % Universidade de S\~ao Paulo \cr \+Cx.\ Postal 20570 --- 01452-990 S\~ao Paulo SP --- Brasil \cr \+{\tt} \cr \vskip 3truemm \+ Instituto de Matematica Pura e Aplicada \cr \+ Estrada Dona Castorina 110 --- 22.460 Jardim Botanico --- Brasil\cr \+ and URA CNRS 1378 --- Universit\'e de Rouen \cr \+ BP 118 --- 76.821 Monts Saint Aignan Cedex --- France \cr \+ {\tt }\cr \bye