Random Walk on the Simple Symmetric Exclusion Process

Augusto Teixeira,Daniel Kious,Marcelo R Hilário

doi:10.1007/s00220-020-03833-x

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.

Highlights

Over the last decades the study of the long-term behavior of the position of a particle subject to the influence of a random environment has received great attention from the physics and mathematics community
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
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 ρ ∈ [0, 1] of the underlying SSEP

Summary

Introduction

Over the last decades the study of the long-term behavior of the position of a particle subject to the influence of a random environment has received great attention from the physics and mathematics community. The main contribution of this paper is to develop a technique that allows one to prove LLN and CLT for random walks on a class of dynamical random environments that includes the simple symmetric exclusion process (SSEP) and the Poisson cloud of independent simple symmetric random walkers (PCRW) For these two specific models, we use lateral space–time mixing bounds together with a decoupling inequality involving small changes in the density (sprinkling) in order to prove a LLN, i.e. the existence of an asymptotic speed, for all densities ρ ∈ (0, 1) except at most two values denoted ρ− and ρ+. They conjecture that it is possible to tune the parameters in order to produce regimes in which the fluctuations of the walker around its limiting speed scale super or sub-diffusively This phenomenon, should be regarded as a manifestation of the strong space–time correlations of the environment which allows, for instance, the existence of traps that survive enough time for being relevant in the long-term behavior. This does not hold in higher dimensions or if we allow for long-range jumps of the walker

Mathematical Setting and Main Results

Graphical Construction and Backbone of the Proof

Lateral Decoupling

Upper and Lower Deviations of the Speed