95-41 P.A. Ferrari ,L. R. G. Fontes
Poissonian Approximation for the Tagged Particle in Asymmetric Simple Exclusion (31K, LaTeX) 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 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.

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