Biased random walk on dynamical percolation

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.

We consider dynamical percolation on the complete graph $K_n$, where each edge refreshes its state at rate $\mu \ll 1/n$, and is then declared open with probability $p = \lambda/n$ where $\lambda > 1$. We study a random walk on this dynamical environment which jumps at rate $1/n$ along every open edge. We show that the mixing time of the full system exhibits cutoff at $\log n/\mu$. We do this by showing that the random walk component mixes faster than the environment process; along the way, we control the time it takes for the walk to become isolated.

Exceptional times for the dynamical discrete web