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

TL;DR

This paper investigates the dynamics of a random walk on a graph with edges that randomly open and close, analyzing mixing times, displacement, and hitting times, revealing behaviors similar to classical lattice models.

Contribution

It provides new results on mixing times, displacement, and hitting times for random walks on dynamical percolation, extending classical lattice results to this dynamic setting.

Findings

Mixing times are proportional to n^2/μ in the subcritical regime.

Mean squared displacement and hitting times are characterized.

Recurrence and transience properties mirror those of Z^d lattice.

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 Z^d holds for this model as well.