Mixing and relaxation time for random walk on wreath product graphs∗

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 o G, the graph whose vertices consist of pairs (f, x) where f = (fv)v∈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 fx and fy according to the transition probabilities on H; fz for z 6= 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.