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\begin{document}
\baselineskip=24pt
\title{Poissonian Approximation for the Tagged Particle in Asymmetric
Simple Exclusion}
\author{{P.A. Ferrari \hspace{10 mm} L. R. G. Fontes}\\
Instituto de Matem\' atica e Estat\'\i stica,\\
Universidade de S\~ao Paulo }
\maketitle
\begin{abstract}
We consider the position of a tagged particle in the one dimensional
asymmetric nearest neighbors simple exclusion process. Each particle attempts
to jump to the site to its right at rate $p$ and to the site to its left at
rate $q$. The jump is realized if the destination site is empty. We assume
$p>q$. The initial distribution is the product measure with density $\lambda$,
conditioned to have a particle at the origin. We call $X_t$ the position
at time $t$ of this particle. Using a result recently proved by the authors
for a semi-infinite zero range process, it is shown that for all $t\ge 0$,
$X_t= N_t -B_t+B_0$, where $\{N_t\}$ is a Poisson process of parameter
$(p-q)(1-\lambda)$ and $\{B_t\}$ is a stationary process satisfying $E\exp
(\theta \vert B_t\vert)<\infty$ for some $\theta>0$. As a corollary we
obtain that ---properly centered and rescaled--- the process $\{X_t\}$
converges to Brownian motion. A previous result says that in the scale
$t^{1/2}$, the position $X_t$ is given by the initial number of empty sites
in the interval $(0,\lambda t)$ divided by $\lambda$. We use this to compute the
asymptotic covariance at time $t$ of two tagged particles initially at sites
$0$ and $rt$. The results also hold for the net flux between two queues in a
system of infinitely many queues in series.
\end{abstract}
\newtheorem{teo}{Theorem}
\newtheorem{lemma}{Lemma}
\newtheorem{rem}{Remark}
\newtheorem{prop}{Proposition}
\newtheorem{cor}{Corollary}
\renewcommand{\a}{\alpha}
\renewcommand{\b}{\beta}
\renewcommand{\O}{\Omega}
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\renewcommand{\n}{\mbox{I}\!\mbox{N}}
\renewcommand{\L}{{\bf L}}
\newcommand{\ls}{{\bf L}^\ast_2}
\newcommand{\la}{\lambda}
\newcommand{\nur}{\nu_{\la}}
\newcommand{\mur}{\mu_{\la}}
\newcommand{\nut}{\nu_2}
\newcommand{\ga}{\gamma}
\newcommand{\ze}{\zeta}
\newcommand{\si}{\sigma}
\newcommand{\sit}{\si_t}
\newcommand{\siz}{\si_0}
\newcommand{\six}{\si^{xy}}
\newcommand{\siy}{\si^{yx}}
\renewcommand{\ss}{\si^\ast}
\newcommand{\sst}{\ss_t}
\newcommand{\sxs}{(\sst,\xst)}
\newcommand{\sx}{(\sit,\xit)}
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\newcommand{\abst}{|\sst|}
\newcommand{\asz}{|\ss_0|}
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\newcommand{\xiy}{\xi^{yx}}
\newcommand{\xs}{\xi^\ast}
\newcommand{\xst}{\xs_t}
\newcommand{\xsz}{\xs_0}
\newcommand{\q}{q^\ast}
\newcommand{\ps}{q^*_b}
\newcommand{\px}{q^*_r}
\newcommand{\qb}{q_b}
\newcommand{\qr}{q_r}
\newcommand{\mus}{\mu^\ast}
\newcommand{\Xst}{X^\ast_t}
\newcommand{\yt}{Y_t}
\newcommand{\yst}{Y^\ast_t}
\newcommand{\at}{A_t}
\newcommand{\Ast}{A^\ast_t}
\newcommand{\dt}{D_t}
\newcommand{\dnt}{D^1_t}
\newcommand{\dst}{D^\ast_t}
\newcommand{\zt}{Z_t}
\newcommand{\zst}{Z^\ast_t}
\newcommand{\fsx}{\frac{\si(x)}{\si(x)+\xi(x)}}
\newcommand{\fsy}{\frac{\si(y)}{\si(y)+\xi(y)}}
\newcommand{\fxx}{\frac{\xi(x)}{\si(x)+\xi(x)}}
\newcommand{\fxy}{\frac{\xi(y)}{\si(y)+\xi(y)}}
\newcommand{\fsu}{\frac{\si(-1)}{\si(-1)+\xi(-1)}}
\newcommand{\fxu}{\frac{\xi(-1)}{\si(-1)+\xi(-1)}}
\newcommand{\black}{$\mbox{\rm black}^\ast\,$}
\newcommand{\red}{$\mbox{\rm red}^\ast\,$}
\newcommand{\beq}{\begin{equation}}
\newcommand{\bteo}{\begin{teo}}
\newcommand{\bcor}{\begin{cor}}
\newcommand{\ecor}{\end{cor}}
\newcommand{\brem}{\begin{rem}}
\newcommand{\erem}{\end{rem}}
\newcommand{\eteo}{\end{teo}}
\newcommand{\eeq}{\end{equation}}
\newcommand{\beqn}{\begin{eqnarray}}
\newcommand{\eeqn}{\end{eqnarray}}
\newcommand{\beqnn}{\begin{eqnarray*}}
\newcommand{\eeqnn}{\end{eqnarray*}}
\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.
\def\ie{{\it i.e.\/}}
\newcommand{\bbz}{{Z\kern-0.45emZ}}
\newcommand{\bbzz}{{Z\kern-0.30emZ}}
\newcommand{\murp}{\mu_{\rho}^{\prime\prime}}
\def\E{\bf E}
\def\xt{X_t}
\def\nt{N_t}
\def\bt{B_t}
\def\bz{B_0}
\def\ra{(p-q)(1-\la)}
\def\zet{\zeta_t}
\def\rzx{\rho_0(x)}
\def\zzt{\zeta^0_t}
\def\zot{\zeta^1_t}
\def\rox{\rho_1(x)}
\def\zez{\zeta_0}
\def\zzz{\zeta^0_0}
\def\zoz{\zeta^1_0}
\def\xzt{Z^0_t}
\def\xot{Z^1_t}
\def\rzt{N^0_t}
\def\rot{N^1_t}
\def\bzt{A^0_t}
\def\bot{A^1_t}
\def\bzz{A^0_0}
\def\boz{A^1_0}
\def\dzt{D^0_t}
\def\dut{D^1_t}
\def\ez{\eta_0}
\def\yt{Y_t}
\def\nb{\bar\nu}
\def\nbr{\bar\nu_\la}
\def\nsr{\nu^\ast_\la}
\def\nsrz{\nu^\ast_{\la_0}}
\def\nsru{\nu^\ast_{\la_1}}
\def\lz{\lambda_0}
\def\lu{\lambda_1}
\def\a{\alpha}
\def\txt{\tilde X^\epsilon_t}
\def\rt{N_t}
\def\epi{\ep^{-1}}
\def\epd{\ep^{2}}
\def\eph{\ep^{1/2}}
\def\epha{\ep^{-1/2}}
\def\xet{X_{\ep^{-1}t}}
\def\xes{X_{\ep^{-1}s}}
\def\ret{\ep^{1/2}R_{\ep^{-1}t}}
\def\bet{\vert B_{\ep^{-1}t}\vert}
\def\beti{B_{\ep^{-1}t_i}}
\def\ep{\epsilon}
\def\ti{t_i}
\def\ii{I_i}
\def\ai{A_i}
\def\d{\delta}
\def\ht{\frac{1}{t}}
\def\lf{\lfloor}
\def\rf{\rfloor}
\def\rtf{\lf rt\rf}
\newcommand{\murl}{\mu_\la^\prime}
\def\rat{(p-q)\la}
\def\nz{n_0\left(\eta,\rat t\right)}
\def\nr{n_{\rtf}\left(\eta,\rat t\right)}
\def\nzf{\frac{n_0\left(\eta_0,\rat t\right)}{\la}}
\def\nrf{\frac{n_{\rtf}\left(\eta,\rat t\right)}{\la}}
\def\nze{n_z(\eta,y)}
\def\one{{\bf 1}}
\vskip 3truemm
\noindent {\it Keywords and phrases.}
Asymmetric simple exclusion, tagged particle, Poissonian approximation, zero
range process, system of queues in series
\vskip 2truemm \noindent {\it AMS-MOS 1991 Classification.} 60K35, 60K25, 90B22,
90B15.
\vskip 2truemm \noindent {\it Running head. Poissonian Approximation}
.
\vskip 3truemm
\noindent {\bf 1. Introduction.}
Let $\et$ be an asymmetric simple exclusion process in $\{0,1\}^\z$ with jump
rates $p$ and $q$ to the right and left respectively, $p>q\geq 0$, starting
with the equilibrium measure $\mur^\prime$, the product of Bernoulli
distributions in $\z$ with constant parameter $\la$ at all sites except at the
origin, where it is 1, and let $\xt$ denote the position of the tagged
particle initially put at the origin.
When the process is totally asymmetric ($q=0$), it is known that $\xt$ is a
Poisson process (Spitzer (1970), Liggett (1985), Kipnis (1986)). This is a
consequence of a theorem of Burke (1956) for queues in series; see Kelly
(1979). We study the case $p>q>0$. Since the system as seen from $X_t$ is in
equilibrium, the expected value of the position of the tagged particle at time
$t$ can be computed immediately: $E\xt =(p-q)(1-\la)t$. Arratia (1983)
conjectured that the asymptotic variance should be the same as the mean:
$$
\lim_{t\to\infty} {E(\xt- (p-q)(1-\la)t)^2\over t} = (p-q)(1-\la).
$$
This was then proven by De Masi and Ferrari (1986). Since then, this identity
between the asymptotic variance and the mean has puzzled us. In this paper we
show that $\xt$ can be approximated by a Poisson process so sharply that the
error can be dominated by a random variable with an exponential tail uniformly
in time. We have two proofs of this, both based on Ferrari and Fontes (1994b)
where we study a zero range process on $\{0,1,\dots\}$ with a sink-source of
customers at $-1$ and with a positive average net output of customers. The
zero range process is also called Jackson network, due to the early work of
Jackson (1963).
We say that a random variable $W$ has a finite exponential moment if there is
a positive constant $\theta$ such that $Ee^{\theta \vert W\vert}$ is finite.
We say that a process $W_t$ has a bounded exponential moment if there is a
positive constant $\theta$ such that $Ee^{\theta\vert W_t\vert}$ is bounded
uniformly in $t$.
Our main results are Theorem 1 and 3 below.
Theorem 1 says that we can write for all $t\ge 0$,
\beq
\label{eq:1111}
\xt = N_t - B_t + B_0
\eeq
where $N_t$ is a Poisson process of parameter $(p-q)(1-\la)$ and $B_t$ is a
stationary process with a bounded exponential moment. To show this result we
use a coupling introduced by Ferrari, Kipnis and Saada (1991) between a semi
infinite process --with a leftmost particle-- and the infinite process. The
semi infinite process as seen from the leftmost particle is isomorphic to the
zero range process mentioned above. The position of the leftmost particle is
represented in this system by the net output of customers. The net output
of customers is known to approach a Poisson process in the sense
(\ref{eq:1111}). Under the coupling, the distance between the tagged particle
and the leftmost particle is proven to have a stationary distribution with an
exponential tail.
In Theorem 3 we show that there exist Poisson processes $\rzt$ and $\rot$
with the same rate $\ra$ and processes $\dzt$ and $\dnt$ in $\bbz$ with
bounded exponential moments such that for all $t\ge 0$,
$$
\rot+\dnt\leq\xt\leq\rzt+\dzt.
$$
This result is contained in Theorem 1. We decided to include it anyway
because
we have an easier and more
general proof. One of the advantages of this proof is that it
can be extended to simple exclusion processes for which jump rates of
different particles can be different. See Benjamini, Ferrari and Landim
(1994). An application to the zero range process is discussed in the last section.
Kipnis (1986) proved a central limit theorem for the tagged particle.
G\"artner and Presutti (1990),
Ferrari (1992) and Ferrari and Fontes (1994a) established that in the scale
$\sqrt t$, the fluctuations of the tagged particle at time $t$ can be read in
the initial configuration. More precisely, let $n_0(\eta,(p-q)\la t)$ be the
number of empty sites of the configuration $\eta$ in the interval
$[0,(p-q)\la t]$. Then, under initial distribution $\nu'_\la$,
\beq
\label{eq:1}
\lim_{t\to\infty}\ht E\left(\xt-\nzf\right)^2=0
\eeq
This is sufficient
to give weak convergence of the finite dimensional distributions to
Brownian Motion, but not tightness. We show in Corollary 1 below
that the tightness is a consequence of the
sharpness of the approximation to a Poisson process.
Kipnis (1986)
proved that two particles initially at distance $\sqrt t$ have exactly the same
asymptotic fluctuations, i.e. that the covariance is the same as the variance
of one of them. Using (\ref{eq:1}) we compute, in Theorem 4 below,
the asymptotic (as $t\to\infty$) covariance at time $t$ of two particles
initially put at the origin and at site (integer part of) $rt$.
\vskip 3truemm
\noindent {\bf 2. Poissonian Approximation}
\bteo
\label{teo:app2}
Let $X_t$ be the position of a tagged particle initially at the origin for the
asymmetric simple exclusion process under initial distribution $\mu'_\la$, the
product measure with density $\la$ conditioned to have a particle at the origin.
There exist a Poisson process $N_t$ with
rate $\ra$ and a stationary processes $B_t$ in $\bbz$ with
bounded exponential
moment such that for all $t\ge 0$,
$$\xt=N_t-B_t+B_0.$$
\eteo
\noindent {\bf Proof.}
This proof is based on a coupling argument introduced by
Ferrari, Kipnis and Saada (1991) which decomposes $\et$ into a first and
second class
particles system evolving in $\z$ as follows. Let $\ez$ be
the initial configuration of $\et$ chosen according to
$\mur^\prime$. Let $x_0=0$ and denote $0x_{-2}>x_{-1}>0$ the positions of the
negative particles.
The particle $x_i$, $i\in\z$, is then labeled ``first class" independently
of the other particles with probability $(p/q)^i/(1+(p/q)^i)$ or
``second class" with the complementary probability. These particles
move like ordinary particles except that when a first class particle
attempts a jump over a second class one, they exchange positions. Call $\ga_t$
and $\ze_t$ the configurations of first and second class particles
respectively. Clearly we can couple the three processes in such a way that
for all $t\ge0$ we have $\ga_t+\ze_t = \eta_t$ coordinatewise. So that, under this
coupling when we disregard the labels we have the doubly infinite system
$\eta_t$ whose particles interact by exclusion. The position $X_t$ denotes
the position of
the $\eta$ tagged particle initially at the origin. When we add the labels, it is
possible that the label of $X_t$ changes with time. At time zero, $X_0$ is
labeled $\ga$ (or first class) with probability $1/2$.
Proposition 3.14 in the above mentioned paper states that this initial
measure for the positions of the particles plus the labels, denoted $\nb$
is invariant for the coupled process
as seen from the $\eta$ tagged particle $\tau_{X_t}(\ga_t,\ze_t)$. The key
point to show this invariance is that the measure $\nb$ is reversible with
respect to the part of the generator that governs
the exchanges between first and second class
particles.
It is clear that under $\nb$,
there is a leftmost first class particle in the system, let us denote by
$\yt$ its (absolute) position at time $t$. Under the invariant measure $\nb$,
the distribution of
\beq
\label{eq:ct}
C_t=\xt - Y_t
\eeq
is independent of $t$ and a direct
computation shows that $C_t$ has a bounded exponential moment.
The process $\yt$ can be studied by seeing $\ga_t$ as a semi infinite system
of queues or a zero range process (Andjel (1982), Jackson (1963)). This
representation was used by Kipnis (1986), Ferrari~(1986), Ferrari and De
Masi~(1985) and De Masi et al. (1988). The queues are indexed by $\{-1,0,$
$1,\ldots\}$ and the number of customers in queue $i\in\{0,1,\ldots\}$ is
given by the number of holes between the $i$-th and $(i+1)$-th $\ga$
particles; the queue at $-1$ having an infinite number of customers.
$Z_t=\yt-Y_0$ can then be represented by the net output of customers from
queue $0$ to queue $-1$. More formally, let $\xi_t$ be the process defined by
$\xi_t(i)=$ number of sites between the $i$th and $(i+1)$-th $\ga_t$
particles. It is easy to see that $\ga_t$ is Markovian, as well as the
process as seen from the leftmost particle $\tau_{Y_t}\ga_t$. The invariance
property of $\nb$ implies that the $\ga$ marginal of $\nb$ as seen from the
leftmost $\ga$ particle is invariant for $\tau_{Y_t}\ga_t$, the $\ga$ process
as seen from the leftmost $\ga$ particle. This implies that if we denote
$\nbr$ the measure induced by the $\ga$ marginal on the semi-infinite zero
range process, then $\nbr$ is invariant for $\xi_t$.
We state now Theorem 2 of Ferrari and Fontes (1994b).
\begin{teo} (Ferrari and Fontes (1994b)) Let $q(x,y)$
be the transition rates of a continuous time positive
recurrent ergodic Markov
process on $\{0,1,\dots\}$ satisfying
\beq
\label{eq:ex}\sup_y\sum_xq(x,y)<\infty,\; \sup_y\sum_xq(y,x)<\infty;
\eeq
Assume that there exists a finite measure $m$ on
$\{0,1,\dots\}$ such that
\beqn
\label{eq:m}
&\displaystyle{\sum_{z\ge -1} q(y,z)m(y) = \sum_{x\ge -1} m(x) q(x,y),\ \ y\ge -1,}\\
\label{eq:m1} &m(-1)=1> m(x),\ x\ge 0\\
\label{eq:ex0}&\displaystyle{\sup_y\, m(y)\sum_{x\ge -1}
\frac{q(y,x)}{m(x)}}<\infty,
\eeqn
Assume that the set of $\phi>0$ for which there exists a sigma finite measure
$\rho=\rho_\phi$ on $\{0,1,\dots\}$ satisfying
\beqn
\label{eq:ro}
\sum_{z\ge -1} q(y,z)\rho(y) = \sum_{x\ge -1} \rho(x) q(x,y),\ \ y\ge 0,\\
\label{eq:ro1}\rho(-1)=1>\rho(x)>m(x), \ x\ge 0;\\
\label{eq:la} \phi = \sum_{x\ge 0} (\rho(x) q(x,-1)-q(-1,x))
\eeqn
is not empty. Fix a $\phi$ in this set and assume that $\rho=\rho_\phi$ satisfies
\beqn
\label{eq:mx}
&\displaystyle{\sum_{x\ge 0} \frac{m(x)}{1-\rho(x)+m(x)}}<\infty;\\
%\eeq
%\item Furthermore
%\beqn
\label{eq:ex1}&\,\displaystyle{\sup_y (\rho(y)-m(y))\sum_{x\ge -1}
\frac{q(y,x)}{\rho(x)-m(x)}}<\infty.
\eeqn
Let $\xi_t$ be the system of infinitely many queues on ${\n}^{\n}$
with rates $q(x,y)$. Let the initial
distribution be the invariant measure
$\nu^*$ defined
for any
finite set $A\subset {\n}$, and any non negative numbers $k(x)\ge 0$, $x\in
A$, by
\beq
\label{eq:nsr}
\nu^*(\xi(x)=k(x),x\in A)=\prod_{x\in A}\rho(x)^{k(x)}(1-\rho(x)),
\eeq
Then for all $t\ge 0$, the net output process $Z_t$ can be
expressed as the following sum
\beq
\label{eq:at}
Z_t = N_t - A_t + A_0
\eeq
where $N_t$ is a Poisson process with rate $\phi$
and $A_t$ is a stationary process on $\n$ with a bounded exponential moment.
\end{teo}
We apply Theorem~2 to the system of queues with
$q(x,y) = p \one\{y=x-1\} +q \one\{y=x+1\}$, $x\ge 0$,
$q(-1,0)=q$ and $\phi=(p-q)(1-\la)$.
In this case $m(x) = (q/p)^{x+1}$ and $\rho(x) = m(x)+(1-m(x))(1-\la)$. We
call $\nsr$ the measure defined in (\ref{eq:nsr}) for this choice of $\rho(.)$.
Recalling that $C_t = X_t - Y_t$, $X_0=0$ and $Z_t = Y_t - Y_0$, we can write
$$
\xt = Z_t + C_t - C_0
$$
Theorem~2 together with the invariance of $\nbr$ and the fact that under it
the distribution of the distance $C_t$ between $\yt$ and $\xt$ is independent of
time and with a finite
exponential moment will conclude the proof by taking $B_t=A_t-C_t$.
To use this result, one has to verify the hypotheses
about the invariant measure $\nbr$.
We have two invariant measures,
$\nbr$ and $\nsr$, the later fitting in the hypotheses
where we would like to fit $\nbr$. It is then sufficient that the two
measures be shown to be the same. The reader could try to show this by
a direct computation.
We found easier to use an indirect approach already present in Ferrari (1986),
inspired in
Liggett (1976) and Andjel (1982). First consider modifications $\nsrz$ and
$\nsru$ of $\nsr$ where
\beq
\label{eq:mon}
\la_0>\la>\la_1.
\eeq
It is clear by the
properties of $\nbr$ (inherited from the product nature of $\nb$) and the
product nature of
$\nsrz$ and $\nsru$ that these measures
have an asymptotic average queue size, respectively,
$(1-\la)/\la$, $(1-\la_0)/\la_0$ and $(1-\la_1)/\la_1$:
\beq
\label{eq:aver}
\lim_{n\to\infty} {1 \over n+1} \sum_{i=0}^n \xi(i) =
{1-\la \over \la} \ \ \ \ \nbr \hbox{ and } \nsr \hbox { almost surely}
\eeq
Ferrari (1986) says that there exists
a coupling $\mu_0$ of $\nbr$ and $\nsrz$ such that if $\xi$ denotes the first
marginal and $\xi^0$ the second,
$$
\mu_0(\xi^0\geq\xi\,\mbox{or}\,\xi^0\leq\xi)=1
$$
and a coupling $\mu_1$ of $\nbr$ and $\nsru$ such that
$$
\mu_1(\xi^1\geq\xi\,\mbox{or}\,\xi^1\leq\xi)=1,
$$
where $\xi,\xi^0$ and $\xi^1$ are the $\nbr,\nsrz$ and $\nsru$
coordinates, respectively. It is clear by (\ref{eq:mon}) and
the asymptotic averages (\ref{eq:aver})
that actually
$\mu_0(\xi^0\leq\xi)=1$ and $\mu_1(\xi^1\geq\xi)=1$.
Thus, $\nsrz\leq\nbr\leq\nsru$ stochastically.
If we replace $\nbr$ by $\nsr$ in this last expression, it clearly
still holds. The equality of $\nbr$ and $\nsr$ follows by letting
$\la_0$ and $\la_1$ tend to $\la$ and the
proof is complete.\square
\bteo
\label{teo:app} Let $X_t$ be as in Theorem 1.
Then there exist Poisson processes $\rzt$ and $\rot$ with the same
rate $\ra$ and processes $\dzt$ and $\dnt$ in $\bbz$ with
bounded exponential
moments such that for all $t\ge 0$,
$$
\rot+\dnt\leq\xt\leq\rzt+\dzt.
$$
\eteo
\noindent {\bf Proof.}
As in the previous proof, the process $\xt$ can be represented as the net flux of
customers between the queues $0$ and $-1$ for a system of infinitely many
queues indexed by $\z$. The queuing system is the zero range process on
$\n^\z$ for which at rate $q$ and $p$ respectively, one customer at queue $i$
jumps to the
right and left nearest neighbor queue respectively. We consider this system
starting with the
invariant measure $\nur$, a product of geometrics indexed by $\z$ with
constant parameter $1-\la$: $\nur(\ze(i)>0) = 1-\la$.
Let us denote this process by $\zet$. Consider
the auxiliary zero range processes $\zzt$ and $\zot$ in
$\n^{\{-1,0,1,\ldots\}}$ with jump rates identical to those of $\zet$ except
that in the first one there are no jumps from $-1$ to 0 (that is, the
corresponding rate $q(-1,0)= 0$) and the second one has infinitely many customers at
$-1$ (that is, the
corresponding rate is $q(-1,0)= q$ independently of the configuration).
Their invariant measures are product of geometrics in each site with
parameters (the probability of at least one customer in site $x\ge 0$)
\beqnn
\rzx=(1-m(x))(1-\la)\\
\rox=m(x)+(1-m(x))(1-\la),
\eeqnn
respectively, where
$m(x)=(q/p)^{x+1}$.
We couple $\zzt$, $\zet$ and $\zot$ in such a way that at time zero
$\zzz(x)\leq\zez(x)\leq\zoz(x)$ for each $x\geq-1$, using the same
``jump arrows" for the three of them (except that those
from $-1$ to 0 are deleted for the first process). This is the basic coupling
for the zero range process, see Andjel (1982).
Let $\xzt$ and $\xot$ denote the net outputs of $\zzt$ and $\zot$,
respectively. That is the net flux of customers between queue $0$ and ``queue
$-1$''. By Theorem~2 we have the
following representation:
\beqnn
\xzt=\rzt-\bzt+\bzz,\\
\xot=\rot-\bot+\boz,
\eeqnn
where $\rzt$ and $\rot$ are Poisson processes with the same rate
$\ra$ and $\bzt$ and $\bot$ are stationary processes on $\n$ with
bounded exponential moments.
The following inequalities hold.
\beqn
\xt\leq\xzt+C^0,\\
\xot\leq\xt+C^1,
\eeqn
where $C^0=\sum_{x\geq0}(\zez(x)-\zzz(x))$ and
$C^1=\sum_{x\geq0}(\zoz(x)-\zez(x))$ are finite random
variables with finite exponential moments. To show the second inequality notice
that since $\zez(x)\leq\zoz(x)$, each time that a $\ze$ customer
jumps from $0$ to $-1$, a $\ze^1$ customer accompanies it. Now assume that a
$\ze^1$ customer exits without a $\ze$ companion. This may be for two
reasons: either it was a $\ze^1$ customer present at time zero, and this is
taken account with the term $C^1$ or it is a customer that entered the system
without a $\ze$ companion after time zero, but in this case the net flux
produced by this customer is null. On the other hand, each time that a $\zeta$
customer jumps from $-1$ to $0$, a $\zeta^1$ customer accompanies it.
The first identity is shown in the same
manner.
We get the result by
identifying $\dzt$ and $\dnt$ respectively with $C^0-\bzt+\bzz$
and $-C^1-\bot+\boz$.\square
\vskip 2truemm
\bcor
Let $\txt=\left(X_{\ep^{-1}t}-\ra\epi t\right)/\sqrt{\ra\epi t}.$ Then $\txt$ converges
weakly to standard Brownian Motion as $\ep\to 0$.
\ecor
\noindent {\bf Proof.}
The result follows from from the weak convergence of the standardized
Poisson process $\ret$ to Brownian motion and the following. Let $B_t$ be the
stationary process of Theorem 1. For any $T>0$,
$$
\sup_{0\leq t\leq T}\eph\bet\to 0
$$
as $\ep\to0$ in probability.
To establish the latter, we argue as follows. Partition $[0,T]$
in intervals of equal lengths $\ep$ (except possibly the last one
which will have length at most $\ep$) and enumerate them
$\ii$ and its extremes
$\ti$ for $i=1,2,\ldots,N$. Notice that $N$ is of the order of $\epi$.
Now notice that $\bet$ will change more than once in $\ii$
(denote this event by $\ai$) only
if there are at least two arrows between the sites ``0" and ``$-1$"
in an interval of length $\ep$
of the zero range representation of $\xt$. This has probability
of the order of $\epd$. Thus, given $\d>0$,
\beqnn
P(\sup_{0\leq t\leq T}\bet>\epha\d)&=&
P(\sup_{1\leq i\leq N}\sup_{t\in\ii}\bet>\epha\d)\\
&\leq&\sum_{i=1}^NP(\sup_{t\in\ii}\bet>\epha\d)\\
&\leq&\sum_{i=1}^NP(\beti>\epha\d-1)+\sum_{i=1}^NP(\ai).
\eeqnn
The first sum is bounded above by $N\times\exp(-\theta\epha\d)$
with a positive constant $\theta$ given by the exponential boundedness
of $\bet$. The second sum is bounded above by $N\times O(\epd)$.
Letting $\ep\to0$, we get the result.\square
\vskip 3truemm
\noindent {\bf 3. Two Point Covariance}
In this section we get an asymptotic expression for the covariance of the
positions of two tagged particles placed initially order $t$ apart.
\bteo
\label{teo:cov} Let $r>0$.
Assume that the initial configuration is determined by first picking a
configuration from $\nur$ and then adding two particles: one at the origin and
the other at site $\lf rt\rf$ --if those sites are not already occupied. Let
$\xt$ and $\yt$ be the positions of these two tagged particles at time $t$.
Then
\beq
\label{eq:cov}
\lim_{t\to\infty}\ht\,\mbox{\rm Cov}\,(\xt,\yt)=
\cases{\quad\quad\quad0&, {\rm if} $r\geq\rat$,\cr
(\rat-r)(1-\la)/\la&, {\rm otherwise.}}
\eeq
\eteo
\noindent {\bf Proof.}
Display (1.5) in Ferrari (1992) and remarks thereof establish
that under our initial distribution, the limit (\ref{eq:1}) holds and
\beq
\label{eq:2}
\lim_{t\to\infty}\ht E\left(\yt-\nrf\right)^2=0,
\eeq
where $\nze=\sum_{x=z}^{z+y}(1-\eta(x))$ is the number of empty sites of the
initial configuration $\eta$ between $z$ and $z+y$.
These equations imply that
\beq
\label{eq:cov1}
\lim_{t\to\infty}\ht\mbox{Cov}\,(\xt,\yt)=
\lim_{t\to\infty}\ht\frac{\mbox{Cov}\left(\nz,\nr\right)}{\la^2}
\eeq
When $r\geq\rat$, the intervals of the summations in $\nz$ and $\nr$ are
disjoint, so the product nature of $\nur$ implies that (\ref{eq:cov1})
vanishes. When $r<\rat$, then it is easy to see that (\ref{eq:cov1}) will
reduce to
$$
\lim_{t\to\infty}\ht\,
\frac{\mbox{Var}\,\left(\sum_{x=rt}^{\rat t}(1-\eta(x))\right)}{\la^2},$$
which is equal to $(\rat-r)(1-\la)/\la$.\square
\vskip 3truemm
\noindent {\bf 4. Remarks on applications to networks of queues.}
\vskip 3truemm
Our results can be applied to a system with infinitely many queues in series
with service times with rate $1$ and with routing matrix $q(x,x+1)=q$,
$q(x,x-1)=p$, that is, a customer after service chooses a new queue between
the nearest neighbors with probabilities $q$ to the right and $p$ to the left.
Theorems 1 says that for this system starting with the product measure
$\nur$ defined in the proof of Theorem 3, the net flux of customers to the
right between two neighboring queues can be expressed as a Poisson process
plus an order 1 stationary process. This is because, as discussed in the
proofs of the theorems, the net flux in the zero range process can be
represented as $\xt$ when one transforms this process into an asymmetric
simple exclusion process as seen from a tagged particle.
The rate of the Poisson process is of course
$(p-q)(1-\la)$.
The proof of Theorem~3 can also be applied to a general network of queues
labeled with the integers with arbitrary jumps between queues. It shows a
Poissonian approximation for the net flux of customers between the queue $0$
and the queues with negative labels provided the three conditions below:
(a) The customers can not jump between a positive queue and a negative queue in
neither direction (\ie\/ the queue at the origin is a bottleneck).
(b) There exists an invariant product distribution under which there is a
positive average net flux between the queue $0$ and the queues with negative
labels.
(c) Consider the semi-infinite system obtained by assuming that the negative
labeled queues have infinitely many customers. Assume that for this system
there exists an invariant measure for the non-negative queues concentrating in
configurations with a finite total number of customers; furthermore under this
invariant measure the number of customers has a finite exponential moment. It
is not hard to see that this hypothesis puts the semi-infinite system under
the conditions of Theorem 2 of Ferrari and Fontes (1994b).
Finally, the proof of Theorem~3 can be applied to systems with finitely many
queues indexed $\{-N,\dots,N\}$ for the net flux of customers between queue
$0$ and the ``negative'' queues provided (a) and (b) above. If in the limit
when $N\to\infty$, (c) above holds, then the approach to the Poisson process
is uniform in $N$. See Ferrari and Fontes (1994b) for a discussion of the
finite system with a sink/source of customers at $-1$.
\vskip 5truemm
\noindent {\bf Acknowledgments.}
This work is partially supported by FAPESP,
Projeto Tem\'atico, 90/3918-5 and CNPq.
\vskip 5truemm
%\vfill\eject
\noindent {\bf References.}
\baselineskip 14pt
\parskip 3truemm
\parindent 0pt
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\vskip 7truemm
Instituto de Matem\'atica e Estat\'\i stica --- Universidade de S\~ao Paulo ---
Cx.~Postal 66.281 --- 05389-970 S\~ao Paulo SP --- Brasil
pablo@ime.usp.br, lrenato@ime.usp.br
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