Variable speed symmetric random walk driven by symmetric exclusion

TL;DR

This paper establishes a quenched functional central limit theorem for a one-dimensional symmetric random walk influenced by a symmetric exclusion process, providing explicit quadratic variation limits without requiring invariant measure existence.

Contribution

It introduces a novel approach to derive the quenched limit by directly analyzing quadratic variation, bypassing the need for invariant measure construction.

Findings

Proves a quenched functional CLT for the model.

Provides an explicit formula for the quadratic variation.

Demonstrates the approach's applicability to similar models.

Abstract

We prove a quenched functional central limit theorem for a one-dimensional random walk driven by a simple symmetric exclusion process. This model can be viewed as a special case of the random walk in a balanced random environment, for which the weak quenched limit is constructed as a function of the invariant measure of the environment viewed from the walk. We bypass the need to show the existence of this invariant measure. Instead, we find the limit of the quadratic variation of the walk and give an explicit formula for it.