Mixing times for Random Walks on Finite Lamplighter Groups

Given a finite graph G, a vertex of the lamplighter graph G♦ = Z2 o G consists of a zero-one labeling of the vertices of G, and a marked vertex of G. For transitive G we show that, up to constants, the relaxation time for simple random walk in G♦ is the maximal hitting time for simple random walk in G, while the mixing time in total variation on G♦ is the expected cover time on G. The mixing time in the uniform metric on G♦ admits a sharp threshold, and equals |G| multiplied by the relaxation time on G, up to a factor of log |G|. For Z2 o Zn, the lamplighter group over the discrete two dimensional torus, the relaxation time is of order n log n, the total variation mixing time is of order n log n, and the uniform mixing time is of order n. For Z2 o Zn when d ≥ 3, the relaxation time is of order n, the total variation mixing time is of order n log n, and the uniform mixing time is of order n. In particular, these three quantities are of different orders of magnitude. AMS 2000 subject classifications: Primary: 60J10; Secondary: 60B15.