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