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\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