Random walk on dynamical percolation in Euclidean lattices: separating critical and supercritical regimes

Abstract

We study the random walk on dynamical percolation of ${\mathbb{Z}}^d$ (resp., the two-dimensional triangular lattice $\mathcal{T}$), where each edge (resp., each site) can be either open or closed, refreshing its status at rate $\hat{I}¼\in (0,1/e]$. The random walk moves along open edges in ${\mathbb{Z}}^d$ (resp., open sites in $\mathcal{T}$) at rate $1$. For the critical regime $p=p_c$, we prove the following two results: on $\mathcal{T}$, the mean squared displacement of the random walk from $0$ to $t$ is at most $O(t\hat{I}{¼}^{5/132-\hat{I}\mu })$ for any $\hat{I}\mu >0$; on ${\mathbb{Z}}^d$ with $d\ge 11$, the corresponding upper bound for the mean squared displacement is $O(t\hat{I}{¼}^{1/2}\log (1/\hat{I}¼))$. For the supercritical regime $p>p_c$, we prove that the mean squared displacement on ${\mathbb{Z}}^d$ is at least $ct$ for some $c=c(d)>0$ that does not depend on $\hat{I}¼$.

26 pages, 1 figure