Poissonian approximation for the tagged particle in asymmetric simple exclusion

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 lambda, 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 greater than or equal to 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\B-1]) < infinity 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.