Fluctuations in the Asymmetric Simple Exclusion Process

Abstract

We review recent results about the asymmetric simple exclusion process in Z. In that process particles jump to the right at rate p and to the left at rate q, p + q = 1. At mosts one particle is allowed in each site. We consider the process in equilibrium with invariant measure v ^p, the product measure with density p. The first result is the computation of the the diffusion coefficient of the current of particles through a fixed point. We find D = |p - q|p(1 - p)|1 - 2p|. A law of large numbers and central limit theorems for the rescaled current are also proven. Analogous results hold for the current of particles through a position travelling at a deterministic velocity r. As a corollary we get that the equilibrium density fluctuations at time t are a translation of the fluctuations at time 0. We also show that the current fluctuations at time t are given, in the scale t ^1/2, by the initial density of particles in an interval of length |(p - q)(1 - 2p)|t. The second result is a representation of the position X _t of a tagged particle of the system that allows us to write X _t = N _t + B _t , where N _t is a Poisson process of parameter (1 - p)(p - q) and B _t is a random variable whose absolute value is stochastically bounded above by B, a random variable with distribution independent of t and with a finite exponential moment. This is an extension of Burke’s theorem for queuing theory.