\documentstyle{article}
\begin{document}
\title{Relaxation time of the one-dimensional
symmetric zero range process with constant rate}
\author{A. Galves\thanks{Partially supported by CNPq grant number 301301/79},
H. Guiol\thanks{Supported by FAPESP grant
number 96/04859-9}\\
Instituto de Matem\'atica e Estat\'\i stica,\\
Universidade de S\~ao Paulo,\\
BP 66281, 05315-970 S\~ao Paulo, SP, Brasil,\\
e-mail: galves@ime.usp.br, herve@ime.usp.br
}
\date{April 1997}
\maketitle
\newcommand{\carn}{\hfill\rule{0.25cm}{0.25cm}}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{conjecture}[theorem]{Conjecture}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\abstract{We prove that the one-dimensional symmetric zero range dynamics,
starting
either with a periodic configuration or with a stationary exponential mixing
probability distribution, converges to equilibrium faster than $\log
t/t^{-1/2}$.
\\
{\bf Key words:} Symmetric zero range process. Convergence rate.
\\
{\bf AMS 1991 Classification:} Primary: 60K35, Secondary: 82C22}
\section{Introduction}
In this paper we present a sharp upper bound for the relaxation time to
equilibrium of the one-dimensional symmetric zero range
process with constant rate, starting either from a periodic configuration or
from a stationary exponentially mixing probability distribution.
We prove that the distance between the law of the process at
time $t$ and the invariant measure with the corresponding average number of
particles decreases faster than $\log t/\sqrt{t}.$
The zero range process was introduced by F. Spitzer in \cite{spitzer}. The
one-dimensional symmetric zero rage process with constant rate can be
informally described as follows.
Particles are distributed on ${\bf Z}$. Any site of ${\bf Z}$ may be occupied
by any finite number of particles. Associated to the sites there are
independent exponential clocks with parameter one. Each time a clock
rings on a site, one of the particles on this site jumps to a nearest neighbor
site chosen with probability $1/2$. In case the site is empty nothing happens.
Ergodic properties of the zero range processes were studied by Andjel in
\cite{andjel}. In particular, in this paper it was proven that for the one
dimensional
symmetric case with constant rate the set of extremal invariant measures is
$\{\mu_{\rho}: \rho\in(0,1]\}$, which are product measures with geometric
number of particles, with parameter $\rho$, at each site.
Upper bound for the rate of convergence of spin-flip systems were obtained by
several authors (see Holley \cite{holley}, for instance, for a review for the
Ising model).
However for conservative systems the situation has been less studied. As far
as we know, the only available results are the following.
For the case with infinitely many independent random walks, a paper by Hoffman
and Rosenthal \cite{hofros} shows that with suitable initial configuration the
rate is bounded above $(1/\sqrt{t})^d$ for all $d\geq 1$.
For the case of the symmetric simple
exclusion in any dimension Cancrini and Galves \cite{cangal} proved that
the rate of convergence is bounded above by $(\log t/\sqrt{t})^d$.
Ours is the first result of the rate of convergence to equilibrium of a
zero range process.
We should also mention the recent papers \cite{deuschel}, \cite{berzed1},
\cite{berzed2} and \cite{laquya}, which consider the $L^2$-decay of
correlations for different
types of processes in equilibrium. In particular in \cite{laquya} it is
proven, for a class of zero range processes in equilibrium, that the
correlation in mean square decays faster than $(\log t)^{d+3}/t^{d/2}$.
It should be stressed that these are equilibrium results which do not imply
anything about the convergence rate for a process starting away from
equilibrium.
Our proof has three main ingredients. The first one is Kesten's remark
(\cite{spitzer}, \cite{kipnis}) about the correspondence
between the zero range process and the simple exclusion process as seen from a
tagged particle. The second ingredient is the observation that at each fixed
time the law of the exclusion process as seen from a tagged particle coincides
with the law of the ordinary process given that there is a particle at the
origin, when the process start with a stationary distribution
(cf. \cite{harris}, \cite{porsto} and \cite{ferrari}). The third ingredient is
the upper bound on the rate of convergence to equilibrium of the symmetric
simple exclusion process presented in \cite{cangal}.
This paper is organized as follows. In section 2 we define the zero range
process and state the theorem. In section 3 we recall some basic facts about
the simple exclusion process which are used in the proof. The proof of the
theorem is given in section 4.
\section{Definitions and statement of the theorem}
Let ${\bf Y}={\bf N}^{\bf Z}$ be the state space of the zero range process. If
$\xi$ is an element of ${\bf Y}$ and $x\not= y$ are elements of ${\bf Z}$,
let us denote by $\xi^{x,y}$ the element of ${\bf Y}$ defined as follows
\[
\xi^{x,y}(z)=\left\{
\begin{array}{ll}
\xi(x)-1, &\mbox{ if }z=x,\\
\xi(y)+1, &\mbox{ if }z=y.\\
\xi(z), &\mbox{ otherwise}.
\end{array}\right.
\]
The
generator of the one-dimensional symmetric zero range process with constant
rate is defined by
\[
Lf(\xi)=\sum_{x\in{\bf Z}}{\bf 1}_{\xi(x)\geq 1}
\left[f(\xi^{x,x+1})-f(\xi)\right]
\]
\[
+\sum_{x\in{\bf Z}}{\bf 1}_{\xi(x)\geq 1},
\left[f(\xi^{x,x-1})-f(\xi)\right]
\]
where $f: {\bf Y}\rightarrow{\bf R}$ is any function depending on a finite
number
of coordinates.
For any $\rho\in (0,1]$, the product measure $\mu_{\rho}$ is defined by
\[
\mu_{\rho}(\xi\in{\bf Y}: \xi(x_i)=k_i, i=1,...,n)=\rho^n(1-\rho)^{\Sigma k_i}
\]
for all $n, k_1,...,k_n,\in{\bf N}$, $x_1,...,x_n\in{\bf Z}$.
Let $\mu$ be a stationary probability measure on ${\bf Y}$. Let us denote by
$\varphi$ the average number of particles on site $0$, {\it i.e.}
$\varphi=\sum_{k=1}^\infty k\mu(\xi(0)=k)$. It will be convenient to define an
associate parameter $\rho\in(0,1]$ by
\[
\varphi=\frac {1-\rho} {\rho}.
\]
\begin{definition}
Let $\mu$ be a stationary probability measure on ${\bf Y}$ with $\varphi=\frac
{1-\rho} {\rho}$.
We shall say
that $\mu$ is exponentially mixing, if there exists $C>0$ and
$\gamma>0$, such that for all $n$, $k_0,...,k_{n-1}\in {\bf N}$
\begin{equation}\label{mixing}
\left\vert\mu\left(\xi(i)=k_i,i=0,...,n-1\right)-\rho^n
(1-\rho)^{\Sigma_{i=0}^{n-1}k_i}
\right\vert\leq C e^{-\gamma (n+\Sigma_{i=0}^{n-1} k_i)}.
\end{equation}
We shall call $\gamma$ the mixing coefficient of $\mu$.
\end{definition}
\begin{definition}
We shall call a configuration $\xi\in{\bf Y}$ periodic, if there exists a
positive integer $m$ such that $\xi(x)=\xi(x+km)$ for any integer $k$. The
period of $\xi$ is then the smallest positive integer $m$ for which that
property holds.
For any periodic configuration $\xi$ of period $m$, we define the parameter
$\rho=\rho(\xi)$ as follows
\[
\frac {1-\rho} {\rho}= \frac 1 m \sum_{x=1}^m \xi(x).
\]
\end{definition}
For each $n\geq 1$ and $k_1,...,k_n$ we define
\[
{\bf W}_{k_1,...,k_n}=\left\{\xi:\xi(i)=k_i, i=1,...,n\right\}.
\]
and $N=n+\sum_{i=1}^n k_i$.
\noindent
{\bf Theorem}
{\sl
For any periodic configuration $\xi$ and any stationary $\mu$ satisfying to
the mixing condition (\ref{mixing}), for all $t$ large enough and all $n$ and
for all $k_1,k_2,...,k_n\in{\bf N}$ we have
\begin{equation}
\left\vert P\left\{\xi_t^*\in{\bf
W}_{k_1,...,k_n}\right\}-\rho^n(1-\rho)^{k_1+...+k_n}\right\vert\leq C\ N \frac
{\log t}{\sqrt{t}},
\end{equation}
where $\xi_t^*$ means either the configuration at time $t$ starting from
$\xi$ or from the initial measure $\mu$. The constant $C$ depends either on the
period, or on the mixing constants, and $\rho$ is the parameter corresponding
either to the periodic configuration, or to the initial distribution.
}
\section{Warming up}
Let us first recall some basic facts about the simple exclusion process which
will be needed in the proof.
For some basic properties of the simple exclusion process we
refer the reader to Liggett's chapter VIII \cite{liggett}.
Let
\[
{\bf X}=\{0,1\}^{\bf Z}.
\]
In what follows we shall denote by $({\eta}_t^*)_t$ the
usual simple symmetrical exclusion process on ${\bf X}$ starting either from a
fixed configuration or from an initial measure, indicated by the upper
index.
The generator of the one-dimensional symmetric simple exclusion process is
defined by
\[
Gf(\eta)=\sum_{x\in{\bf Z}} \frac 1 2\left[f(\eta^{x,x+1})-f(\eta)\right]
\]
where $\eta^{x,y}$ is the element of ${\bf X}$ defined as follows
\[
\eta^{x,y}(z)=\left\{
\begin{array}{ll}
\eta(y), &\mbox{ if }z=x,\\
\eta(x), &\mbox{ if }z=y.\\
\eta(z), &\mbox{ otherwise}.
\end{array}\right.
\]
and $f: {\bf X}\rightarrow{\bf R}$ is any function depending on a finite
number of coordinates.
We need also to introduce the dual process. It is a pure jump Markov process
take values in ${\cal Z}$, the set of all finite subsets of ${\bf Z}$.
Its generator is
defined by
\[
\tilde{G}f(A)=\sum_{x\in{\bf Z}} \frac 1 2\left[f(A^{x,x+1})-f(A)\right]
\]
where for any $A\in{\cal Z}$, $A^{x,y}$ is the element of ${\cal Z}$ defined
as
\[
A^{x,y}=A\triangle\{x,y\},
\]
where $\triangle$ stands for the symmetric difference
and $f: {\cal Z}\rightarrow{\bf R}$ is any bounded function.
Let us denote by $Z_t^A$ as the dual process at time $t$ starting from set
$A$. By a notational abuse we shall write $Z_t^a$ instead of $Z_t^{\{a\}}$. We
recall that, using Harris graphical construction, it is possible to construct a
coupled family $\{(Z_t^x)_t,x\in{\bf Z}\}$, in such a way that, for any finite
$A$, the following {\sl additivity} property holds
\[
Z_t^A=\bigcup_{a\in A}Z_t^a.
\]
The symmetric simple exclusion process and its dual are related by the formula
\begin{equation}\label{dudu}
{\bf P}\{\eta_t^{\zeta}(a)=1, a\in A\}={\bf P}\{\zeta(Z_t^a)=1, a\in A\}.
\end{equation}
For more details on duality and graphical representation we refer the reader
to Harris seminal paper \cite{harris2}.
A key ingredient of our proof is the following theorem \cite{cangal}
{\bf Theorem}(Cancrini-Galves (1995))
{\sl
Let $\eta\in{\bf X}$ be any periodic configuration and let $\nu$ be any
stationary distribution satisfying the mixing condition
\begin{equation}\label{mixnua}
\nu(\eta\in{\bf X}:\eta(0)=\eta(x+1)=1)-{\rho}^2\leq C e^{-\bar{\gamma}x},
\end{equation}
for any $x\geq 0$. Then
for all $t$ large enough and all $n\in{\bf N}$ we have
\begin{equation}\label{inecangal}
\left\vert P\left\{\eta_t^*\in{\bf
\Lambda_n}\right\}-\rho^n\right\vert\leq C\ n\frac{\log t}{\sqrt{t}},
\end{equation}
where $\Lambda_n=\{\eta:\eta(x)=1, x=1,...,n\}$ and $\eta_t^*$ denotes the
one-dimensional
symmetric simple exclusion process at time
$t$, starting either from
$\eta$ or from the initial measure $\nu$. The constant $C$ depends either on
the
period, or on the mixing constants, and $\rho$ is the density of the initial
distribution.
}
Actually in \cite{cangal} the result is stated and proved for any dimension
and the mixing condition is stated in a more restrictive way. However in the
one-dimensional case it is clear that the simplified mixing condition
(\ref{mixnu}) is sufficient.
Let us now recall the relation between the one-dimensional nearest-neighbor
simple exclusion and the zero range process with constant rate.
Let
\[
\hat{\bf X}=\{\eta\in{\bf
X}:\eta(0)=1,\sum_{x\geq 0}\eta(x)=\infty,\sum_{x\leq 0}\eta(x)=\infty\}.
\]
and let $(\widehat{\eta}_t^*)_t$ be the simple symmetrical
exclusion process as seen from the tagged particle on $\hat{\bf X}$.
If $\nu$ is translation invariant measure on ${\bf X}$, let us denote by
$\hat{\nu}$ its Palm measure which is a probability measure on $\hat{\bf X}$,
defined by $\hat{\nu}(.)=\nu(.\vert\eta(0)=1)$.
Let us define a map from ${\bf Y}$ to $\hat{\bf X}$ in the following way.
For any $\xi\in{\bf Y}$, let us define $X_0=0$ and for any integer $i\geq 1$
\[
X_i=\sum_{k=0}^{i-1}\xi(k)+i\mbox{ and }
X_{-i}=-\sum_{k=1}^i\xi(k)-i.
\]
Let us define the configuration $\hat{\eta}\in\hat{\bf X}$ corresponding to $\xi$ as
follows
\[
\hat{\eta}(x)=
\left\{\begin{array}{ll}
1, & \mbox{ if }x=X_i\mbox{ for some }i\in{\bf Z};\cr
0, & \mbox{ otherwise}.
\end{array}\right.
\]
Given a stationary probability measure $\mu$ on ${\bf Y}$, let us denote by
$\hat{\nu}$ the image of $\mu$ by this map. This is a notational abuse, as
$\hat{\nu}$ is not define as a Palm measure. Actually, the additional
hypothesis that $\mu$ has an average number of particles at site 0 equal to
$\varphi=\frac {1-\rho} {\rho}$, assure the existence of a probability measure
$\nu$ on ${\bf X}$ with density $\rho$ and such that the above defined
$\hat{\nu}$ is its Palm measure.
Given a one dimensional symmetric zero range process with constant rate
$(\xi_t)_t$, let $(\hat{\eta}_t)_t$ be its image by this map. The process
$(\hat{\eta}_t)_t$ is the symmetric simple exclusion process as seen from a
tagged particle (cf. \cite{ferrari}).
For any stationary measure $\nu$ on ${\bf X}$ and any cylinder $C$, we have
\begin{equation}\label{Pablo}
{\bf P}\left\{\eta_t^{\nu}\in C \vert \eta_t^{\nu}(0)=1 \right\}
={\bf P}\left\{\widehat{\eta}_t^{\widehat{\nu}}\in C \right\},
\end{equation}
(cf \cite{harris}, \cite{porsto} and \cite{ferrari}).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section{Proof of the theorem}
Let us start with the proof of the mixing case. We first prove the result when
the
mixing coefficient $\gamma$ is larger than $\log 2$.
\begin{lemma}
Let $\mu$ be a stationary measure satisfying to the mixing condition
(\ref{mixing}) with parameter $\rho$ and mixing coefficient $\gamma>\log 2$.
Then for all $t$ large enough, all $n$ and
for all $k_1,k_2,...,k_n\in{\bf N}$ we have
\begin{equation}
\left\vert P\left\{\xi_t^{\mu}\in{\bf
W}_{k_1,...,k_n}\right\}-\rho^n(1-\rho)^{k_1+...+k_n}\right\vert\leq C\ N \frac
{\log t}{\sqrt{t}},
\end{equation}
where $\xi_t^{\mu}$ means the configuration at time $t$ starting from
the initial measure $\mu$. The constant $c$ depends on the mixing constants.
\end{lemma}
\noindent
{\bf Proof}
We first remark that the
mixing condition (\ref{mixing}) for the zero range
process implies a good mixing condition for the associated simple exclusion
process.
Let $\hat{\nu}$ be the probability measure on $\hat{\bf X}$ corresponding
to the measure $\mu$ on ${\bf Y}$.
By definition
\[
\hat{\nu}(\hat{\eta}\in\hat{\bf X}:\hat{\eta}(n+1)=1)=
\]
\begin{equation}\label{mixu}
=\mu(\xi\in{\bf Y}:\xi(0)=n)+
\sum_{k=1}^n\mu\left(\bigcup_{1\leq j_1<...0$.
By definition
\[
\hat{\nu}(\hat{\eta}\in\hat{\bf X}:\hat{\eta}(n+1)=1)=\frac
{\nu(\eta\in{\bf X}:\eta(0)=\eta(n+1)=1)} {\nu(\eta\in{\bf X}:\eta(0)=1)}.
\]
By hypothesis $\nu(\eta\in{\bf X}:\eta(0)=1)=\rho$. Therefore, we obtain the
mixing condition
\begin{equation}\label{mixnu}
\nu(\eta\in{\bf X}:\eta(0)=\eta(n+1)=1)-{\rho}^2\leq C e^{-\bar{\gamma}n}.
\end{equation}
As recalled in section 3, condition (\ref{mixnu}) assures that the rate of
convergence of the exclusion process is bounded above by $\log t/\sqrt{t}$
\cite{cangal}.
To apply this result for the zero range we first remark that
\[
{\bf P}\left\{\xi_t^{\mu}\in{\bf W}_{k_1,...,k_n}\right\}=
{\bf P}\left\{\widehat{\eta}_t^{\widehat{\nu}}\in C_{x_0,...,x_n}\right\}
\]
where $x_0=0$,$x_1=k_1+1$,...,$x_n=\sum_{i=1}^n k_i+n$,
\[
C_{x_0,...,x_n}=\{\eta\in{\bf X}:\eta(x_0)=...=\eta(x_n)=1,\eta(x)=0,x\in I_{x_0,...,x_n}\},
\]
and $I_{x_0,...,x_n}=\{y\in{\bf Z}:x_00$, such that for
any stationary measure $\mu$ on ${\bf Y}$, satisfying the mixing condition
(\ref{mixing}) with mixing coefficient $\gamma_0$, the law of
$\xi_{t_0}^{\mu}$ satisfies the mixing condition with mixing coefficient
$\gamma_1$,
where $\xi_t^{\mu}$ stands for the one-dimensional symmetric zero range
process with constant rate starting from the initial measure $\mu$.
\end{lemma}
{\bf Proof}
To avoid tedious details we shall only write the proof for a cylinder of size
two. The general case is done in exactly the same way.
Using again the map from ${\bf Y}$ to $\hat{\bf X}$, for any
$k_1,k_2\in{\bf N}$, we have
\begin{equation}\label{again}
{\bf P}\{\xi_t^{\mu}\in{\bf W}_{k_1,k_2}\}=
{\bf P}\left\{\widehat{\eta}_t^{\widehat{\nu}}\in C_{x_0,x_1,x_2}\right\}
\end{equation}
By (\ref{Pablo})
\[
{\bf P}\left\{\widehat{\eta}_t^{\widehat{\nu}}\in C_{x_0,x_1,x_2}\right\}
=\frac {{\bf P}\left\{\eta_t^{{\nu}}\in C_{x_0,x_1,x_2}\right\}}
{{\bf P}\left\{\eta_t^{\nu}(0)=1\right\}}
\]
By hypothesis, ${\bf P}\left\{\eta_t^{\nu}(0)=1\right\}=\rho$.
By duality
\[
{\bf P}\left\{\eta_t^{{\nu}}\in C_{x_0,x_1,x_2}\right\}
\]
\begin{equation}\label{equdual}
=\sum_{u,v,w}{\bf P}(Z_t^0=u,Z_t^{k_0+1}=v,Z_t^{k_0+k_1+2}=w)
{\bf P}\left\{\eta^{{\nu}}\in C_{u,v,w}\right\},
\end{equation}
where $(Z_t^{x})$ is the dual process starting at $x$.
The right hand side of (\ref{equdual}) can be bonded above by
\[
\leq\sum_{ud}
{\bf P}(Z_t^0=u,Z_t^{k_0+1}=v,Z_t^{k_0+k_1+2}=w)
{\bf P}\left\{\eta^{{\nu}}\in C_{u,v,w}\right\}
\]
\begin{equation}\label{maj}
+{\bf P}(\vert Z_t^0-Z_t^{k_0+k_1+2}\vertd}{\bf
P}(Z_t^0=u,Z_t^{k_0+1}=v,Z_t^{k_0+k_1+2}=w)\times
\]
\[
\left\vert\frac 1 {\rho}
{\bf P}\left\{\eta^{{\nu}}\in C_{u,v,w}\right\}
-\rho^2(1-\rho)^{w-u-2}
\right\vert
\]
\[
+\frac 1 {\rho}{\bf P}(\vert Z_t^0-Z_t^{k_0+k_1+2}\vertk_0+k_1+2$ and $t_0$ such that the second member of
this inequality is bounded by
\[
C' e^{\gamma_1(1+k_0+k_1)}.
\]
\carn
This concludes the proof of the theorem in the mixing case.
Now lets turn to the proof of the periodic case.
If we start the process with a periodic configuration $\xi$
in ${\bf Y}$ with period say $m$ then the corresponding configuration in
$\hat{\bf X}$ will be also periodic with period $m'=m+\xi(0)+...+\xi(m-1)$.
Then again by Cancrini-Galves's theorem
\cite{cangal} the symmetric simple exclusion
process starting with that periodic configuration (with one particle at the
origin) will converge to the equilibrium with speed bounded by $C\log t/\sqrt{t}$.
As at time $t$ there will be a particle at the origin with a probability
\[
\rho- C\frac {\log t} t\leq{\bf P}(\eta^{\eta_0}_t(0)=1)\leq\rho+ C\frac {\log t} t,
\]
so that it will also reach the equilibrium for the exclusion process as seen
from the tagged particle with speed $C'\log t/\sqrt{t}$, where $C'>C$ only
depends on $C$ and $\rho$.
\carn
\noindent
\appendix{\bf Acknowledgments}
This research is part of FAPESP's {\sl Projeto Tem\'atico} 95/0790-1 and Pronex
No 41.96.0923.00.
We thank E. Andjel, V. Belitsky, L. Bertini, N. Cancrini and
C. Landim for interesting discussions.
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