Mixing time for random walk on supercritical dynamical percolation

TL;DR

This paper proves that the mixing time of a random walk on supercritical dynamical percolation on a high-dimensional torus is proportional to n^2 plus the inverse of the refresh rate, under certain conditions.

Contribution

It establishes the conjectured mixing time behavior for supercritical dynamical percolation, extending previous results from subcritical cases and providing a quenched version of the conjecture.

Findings

Mixing time is Θ(n^2 + 1/μ) for supercritical percolation.

The proof uses percolation theory and evolving set processes.

Results depend on the percolation parameter p and the dimension d.

Abstract

We consider dynamical percolation on the $d$-dimensional discrete torus of side length $n$, $\mathbb{Z}_n^d$, where each edge refreshes its status at rate $\mu=\mu_n\le 1/2$ to be open with probability $p$. We study random walk on the torus, where the walker moves at rate $1/(2d)$ along each open edge. In earlier work of two of the authors with A. Stauffer, it was shown that in the subcritical case $p<p_c(\mathbb{Z}^d)$, the (annealed) mixing time of the walk is $\Theta(n^2/\mu)$, and it was conjectured that in the supercritical case $p>p_c(\mathbb{Z}^d)$, the mixing time is $\Theta(n^2+1/\mu)$; here the implied constants depend only on $d$ and $p$. We prove a quenched (and hence annealed) version of this conjecture up to a poly-logarithmic factor under the assumption $\theta(p)>1/2$. Our proof is based on percolation results (e.g., the Grimmett-Marstrand Theorem) and an analysis of the volume-biased evolving set process; the key point is that typically, the evolving set has a substantial intersection with the giant percolation cluster at many times. This allows us to use precise isoperimetric properties of the cluster (due to G. Pete) to infer rapid growth of the evolving set, which in turn yields the upper bound on the mixing time.