Random Walk on the Simple Symmetric Exclusion Process

Abstract

We investigate the long-term behavior of a random walker evolving on top of the simple symmetric exclusion process (SSEP) at equilibrium, in dimension one. At each jump, the random walker is subject to a drift that depends on whether it is sitting on top of a particle or a hole, so that its asymptotic behavior is expected to depend on the density rho in [0, 1] of the underlying SSEP. Our first result is a law of large numbers (LLN) for the random walker for all densities rho except for at most two values rho _-, rho _+ in [0, 1]. The asymptotic speed we obtain in our LLN is a monotone function of rho . Also, rho _- and rho _+ are characterized as the two points at which the speed may jump to (or from) zero. Furthermore, for all the values of densities where the random walk experiences a non-zero speed, we can prove that it satisfies a functional central limit theorem (CLT). For the special case in which the density is 1/2 and the jump distribution on an empty site and on an occupied site are symmetric to each other, we prove a LLN with zero limiting speed. We also prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.