On the Eigenspaces of Lamplighter Random Walks and Percolation Clusters on Graphs

TL;DR

This paper establishes a connection between lamplighter random walks and percolation clusters on graphs, revealing spectral properties and eigenfunction bases in different regimes.

Contribution

It demonstrates that the Plancherel measure of lamplighter walks equals the expected spectral measure of percolation clusters, providing new spectral insights.

Findings

Spectral measure coincidence in lamplighter and percolation models

Pure point spectrum in the subcritical regime

Complete orthonormal basis of finitely supported eigenfunctions

Abstract

We show that the Plancherel measure of the lamplighter random walk on a graph coincides with the expected spectral measure of the absorbing random walk on the Bernoulli percolation clusters. In the subcritical regime the spectrum is pure point and we construct a complete orthonormal basis of finitely supported eigenfunctions.