Random walks on dynamical percolation: mixing times, mean squared displacement and hitting times

Abstract

We study the behavior of random walk on dynamical percolation. In this model, the edges of a graph $$G$$ are either open or closed and refresh their status at rate $$\mu $$ while at the same time a random walker moves on $$G$$ at rate 1 but only along edges which are open. On the $$d$$ -dimensional torus with side length $$n$$ , we prove that in the subcritical regime, the mixing times for both the full system and the random walker are $$n^2/\mu $$ up to constants. We also obtain results concerning mean squared displacement and hitting times. Finally, we show that the usual recurrence transience dichotomy for the lattice $${\mathbb {Z}}^d$$ holds for this model as well.