Mixing time for random walk on supercritical dynamical percolation

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.