Mixing and relaxation time for Random Walk on Wreath Product Graphs

TL;DR

This paper analyzes the mixing and relaxation times of a generalized lamplighter random walk on wreath product graphs, relating these times to properties of the component graphs G and H.

Contribution

It provides new estimates for the mixing and relaxation times of the lamplighter chain on wreath product graphs, connecting these to hitting times and relaxation times of G and H.

Findings

Mixing time depends on parameters of G and H.

Relaxation time is of the same order as the maximal hitting time plus |G| times H's relaxation time.

Explicit bounds for mixing and relaxation times are derived.

Abstract

Suppose that G and H are finite, connected graphs, G regular, X is a lazy random walk on G and Z is a reversible ergodic Markov chain on H. The generalized lamplighter chain X* associated with X and Z is the random walk on the wreath product H\wr G, the graph whose vertices consist of pairs (f,x) where f=(f_v)_{v\in V(G)} is a labeling of the vertices of G by elements of H and x is a vertex in G. In each step, X* moves from a configuration (f,x) by updating x to y using the transition rule of X and then independently updating both f_x and f_y according to the transition probabilities on H; f_z for z different of x,y remains unchanged. We estimate the mixing time of X* in terms of the parameters of H and G. Further, we show that the relaxation time of X* is the same order as the maximal expected hitting time of G plus |G| times the relaxation time of the chain on H.