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

TL;DR

This paper analyzes the behavior of a random walk on dynamical percolation models in Euclidean lattices, establishing bounds on displacement in critical regimes and demonstrating linear growth in supercritical regimes.

Contribution

It provides new bounds on the mean squared displacement of the random walk in critical regimes and confirms linear growth in supercritical regimes, advancing understanding of dynamical percolation.

Findings

On the triangular lattice, displacement is at most O(tμ^{5/132-ε}) in the critical regime.

On Z^d with d≥11, displacement is at most O(tμ^{1/2} log(1/μ)) in the critical regime.

In the supercritical regime, displacement grows linearly with time, at least ct.

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 $\mu\in (0,1/e]$. 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 $\mu$.