Symmetric exclusion as a random environment: invariance principle

TL;DR

This paper proves that a one-dimensional random walk in a dynamic exclusion environment converges to a combination of Brownian motion and a Gaussian process after rescaling, revealing complex limiting behavior.

Contribution

It establishes an invariance principle for a random walk in a speed-change exclusion environment, a novel result in the study of dynamic random environments.

Findings

Random walk converges to a sum of Brownian motion and Gaussian process.

Environment starts from equilibrium and influences the walk's limiting behavior.

Rescaling reveals a mixed Gaussian process as the limit.

Abstract

We establish an invariance principle for a one-dimensional random walk in a dynamical random environment given by a speed-change exclusion process. The jump probabilities of the walk depend on the configuration of the exclusion in a finite box around the walker. The environment starts from equilibrium. After a suitable space-time rescaling, the random walk converges to a sum of two independent processes, a Brownian motion and a Gaussian process with stationary increments.