# A comparison principle for random walk on dynamical percolation

## TL;DR

This paper compares the mixing, hitting times, and spectral properties of random walk on dynamical percolation with simple random walk on various graphs, extending previous results to new graph classes and critical cases.

## Contribution

It provides general comparison results for dynamical percolation models on arbitrary graphs, including extensions to the critical case and specific bounds for transitive graphs and hypercubes.

## Key findings

- Comparison results for hitting and mixing times
- Bounds on spectral-gap and log-Sobolev constants
- Extension of results to critical percolation on tori

## Abstract

We consider the model of random walk on dynamical percolation introduced by Peres, Stauffer and Steif (2015). We obtain comparison results for this model for hitting and mixing times and for the spectral-gap and log-Sobolev constant with the corresponding quantities for simple random walk on the underlying graph $G$, for general graphs. When $G$ is the torus $\mathbb{Z}_n^d$, we recover the results of Peres et al. and we also extend them to the critical case. We also obtain bounds in the cases where $G$ is a transitive graph of moderate growth and also when it is the hypercube.