On the Spectrum of Lamplighter Groups and Percolation Clusters

Let G be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let H be a finite group and H ≀G the lamplighter group (wreath product) over G with group of “lamps” H. We show that the spectral measure (Plancherel measure) of any symmetric “switch–walk–switch” random walk on H ≀G coincides with the expected spectral measure (integrated density of states) of the random walk with absorbing boundary on the cluster of the group identity for Bernoulli site percolation on X with parameter p = 1/|H|. The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilites on the percolation cluster. In particular, if the clusters of percolation with parameter p are almost surely finite then the spectrum of the lamplighter group is pure point. This generalizes results of Grigorchuk and Żuk, resp. Dicks and Schick regarding the case when G is infinite cyclic. Analogous results relate bond percolation with another lamplighter random walk. In general, the integrated density of states of site (or bond) percolation with arbitrary parameter p is always related with the Plancherel measure of a convolution operator by a signed measure on H ≀ G , where H = Z or another suitable group.