Speed of random walk on dynamical percolation in nonamenable transitive graphs

TL;DR

This paper investigates the speed of a random walk on a nonamenable transitive graph under dynamical percolation, revealing different behaviors in subcritical, critical, and supercritical regimes.

Contribution

It provides the first bounds on the random walk speed in dynamical percolation across all regimes on nonamenable transitive graphs.

Findings

Speed is at most O(√μ log(1/μ)) at criticality for μ ≤ e^{-1}

Speed is of order 1 in the supercritical regime regardless of μ

Speed is of order μ in the subcritical regime

Abstract

Let $G$ be a nonamenable transitive unimodular graph. In dynamical percolation, every edge in $G$ refreshes its status at rate $\mu>0$, and following the refresh, each edge is open independently with probability $p$. The random walk traverses $G$ only along open edges, moving at rate $1$. In the critical regime $p=p_c$, we prove that the speed of the random walk is at most $O(\sqrt{\mu \log(1/\mu)})$, provided that $\mu \le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $\mu)$, while in the subcritical regime $p<p_c$, the speed is of order $\mu\wedge 1$.