Mixing time for random walk on supercritical dynamical percolation

Abstract

We consider dynamical percolation on the d-dimensional discrete torus {mathbb {Z}}_n^d of side length n, where each edge refreshes its status at rate mu =mu _nle 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. When theta (p)>0, we prove a version of this conjecture for an alternative notion of mixing time involving randomised stopping times. The latter implies sharp (up to poly-logarithmic factors) upper bounds on exit times of large balls throughout the supercritical regime. Our proofs are 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.