Random walk on the simple symmetric exclusion process

TL;DR

This paper studies the long-term behavior of a random walk on the simple symmetric exclusion process, establishing laws of large numbers and central limit theorems depending on particle density, with special cases analyzed.

Contribution

It provides the first LLN and CLT results for a random walk on SSEP at equilibrium, characterizing the speed as a function of density and identifying critical densities where speed changes.

Findings

LLN holds for all densities except at most two points.

Asymptotic speed is a monotone function of density.

Functional CLT applies when the speed is non-zero.

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.