On the spectrum of lamplighter groups and percolation clusters

Franz Lehner; Markus Neuhauser; Wolfgang Woess

doi:10.1007/s00208-008-0222-7

paper

On the spectrum of lamplighter groups and percolation clusters

arXiv:0712.3135 · doi:10.1007/s00208-008-0222-7

Abstract

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 Zuk, 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.

minor corrections, a somewhat shortened version to appear in Math Ann

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On the spectrum of lamplighter groups and percolation clusters

Abstract

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