Poissonian approximation for the tagged particle in asymmetric simple exclusion

Abstract

We consider the position of a tagged particle in the one-dimensional asymmetric nearest-neighbor 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 λ, conditioned to have a particle at the origin. We call X, 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 ≧ 0, Xt = Nt − Bt + B 0 , where {N t} is a Poisson process of parameter (p – q)(1– λ) and {Bt } is a stationary process satisfying E exp (θ | B, |) < ∞ for some θ > 0. As a corollary we obtain that, properly centered and rescaled, the process {Xt } converges to Brownian motion. A previous result says that in the scale t 1/2, the position Xt is given by the initial number of empty sites in the interval (0, λt) divided by λ. 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.