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. Finally, we prove similar LLN and CLT results for a different environment, given by a family of independent simple symmetric random walks in equilibrium.

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• Mixing time for the asymmetric simple exclusion process in a random environment
• Random walk in a birth-and-death dynamical environment