On the spectrum of lamplighter groups and percolation clusters

TL;DR

This paper establishes a deep connection between the spectral properties of lamplighter groups and percolation clusters, showing how random walks on these groups relate to percolation models and their spectral measures.

Contribution

It generalizes previous results by linking the spectral measure of lamplighter groups to the expected spectral measure of percolation clusters for arbitrary groups and percolation parameters.

Findings

Spectral measure of lamplighter groups matches expected spectral measure of percolation clusters.

Return probabilities of lamplighter walks coincide with annealed return probabilities on clusters.

If percolation clusters are almost surely finite, the spectrum is pure point.

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 $H \wr G$ 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 $H \wr G$ 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 $H \wr G$, where $H = Z$ or another suitable group.