Mixing time of random walk on dynamical random cluster

TL;DR

This paper analyzes the mixing time of a random walk on a dynamical random cluster environment on a torus, revealing it scales as n^2 divided by the edge switching rate μ for small p, using a novel multi-scale coupling method.

Contribution

It introduces a non-Markovian coupling technique with multi-scale analysis to study mixing times in a dynamical random cluster environment, a novel approach in this context.

Findings

Mixing time scales as n^2/μ for small p.

Constructs a non-Markovian coupling method.

Provides insights into environment dynamics and random walk behavior.

Abstract

We study the mixing time of a random walker who moves inside a dynamical random cluster model on the d-dimensional torus of side-length n. In this model, edges switch at rate \mu between open and closed, following a Glauber dynamics for the random cluster model with parameters p,q. At the same time, the walker jumps at rate 1 as a simple random walk on the torus, but is only allowed to traverse open edges. We show that for small enough p the mixing time of the random walker is of order n^2/\mu. In our proof we construct of a non-Markovian coupling through a multi-scale analysis of the environment, which we believe could be more widely applicable.