 9541 P.A. Ferrari ,L. R. G. Fontes
 Poissonian Approximation for the Tagged Particle in Asymmetric
Simple Exclusion
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Feb 6, 95

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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 semiinfinite 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
$(pq)(1\lambda)$ 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.
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