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

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 $\mu\in (0,1/e]$. 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 $O(t\mu^{5/132-\epsilon})$ for any $\epsilon>0$; on $\mathbb{Z}^d$ with $d\geq 11$, the corresponding upper bound for the mean squared displacement is $O(t \mu^{1/2}\log(1/\mu))$. 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 $\mu$.