# 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.

## Key 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.