On the spectrum of lamplighter groups and percolation clusters

Abstract

Let ${\mathfrak{G}}$ be a finitely generated group and X its Cayley graph with respect to a finite, symmetric generating set S. Furthermore, let ${\mathfrak{H}}$ be a finite group and ${\mathfrak{H}\wr\mathfrak{G}}$ the lamplighter group (wreath product) over ${\mathfrak{G}}$ with group of “lamps” ${\mathfrak{H}}$ . We show that the spectral measure (Plancherel measure) of any symmetric “switch–walk–switch” random walk on ${\mathfrak{H}\wr\mathfrak{G}}$ 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 ${\mathsf{p} = 1/|\mathfrak{H}|}$ . The return probabilities of the lamplighter random walk coincide with the expected (annealed) return probabilities on the percolation cluster. In particular, if the clusters of percolation with parameter ${\mathsf{p}}$ 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 ${\mathfrak{G}}$ 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 ${\mathsf{p}}$ is always related with the Plancherel measure of a convolution operator by a signed measure on ${\mathfrak{H}\wr\mathfrak{G}}$ , where ${\mathfrak{H}=\mathbb{Z}}$ or another suitable group.