Uniform Mixing Time for Random Walk on Lamplighter Graphs

Suppose that G is a finite, connected graph and X is a lazy random walk on G. The lamplighter chain X associated with X is the random walk on the wreath product G = Z2 oG, the graph whose vertices consist of pairs (f, x) where f is a labeling of the vertices of G by elements of Z2 and x is a vertex in G. There is an edge between (f, x) and (g, y) in G if and only if x is adjacent to y in G and f(z) = g(z) for all z 6= x, y. In each step, X moves from a configuration (f, x) by updating x to y using the transition rule of X and then sampling both f(x) and f(y) according to the uniform distribution on Z2; f(z) for z 6= x, y remains unchanged. We give matching upper and lower bounds on the uniform mixing time of X provided G satisfies mild hypotheses. In particular, when G is the hypercube Z2, we show that the uniform mixing time of X is Θ(d2). More generally, we show that when G is a torus Zn for d ≥ 3, the uniform mixing time of X is Θ(dn) uniformly in n and d. A critical ingredient for our proof is a concentration estimate for the local time of random walk in a subset of vertices.