Speed of random walk on dynamical percolation in nonamenable transitive graphs

Abstract

Let $G$ be a nonamenable transitive unimodular graph. In dynamical percolation, every edge in $G$ refreshes its status at rate $\hat{I}¼>0$, 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 $O(\sqrt{\hat{I}¼\log (1/\hat{I}¼)})$, provided that $\hat{I}¼\le e^{-1}$. In the supercritical regime $p>p_c$, we prove that the speed on $G$ is of order 1 (uniformly in $\hat{I}¼)$, while in the subcritical regime $p<p_c$, the speed is of order $\hat{I}¼\wedge 1$.

29 pages, 1 figure