On the eigenspaces of lamplighter random walks and percolation clusters on graphs

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 set of finitely supported eigenfunctions.

Random walks on supercritical percolation clusters

We obtain Gaussian upper and lower bounds on the transition density qt(x, y) of the continuous time simple random walk on a supercritical percolation cluster C∞ in the Euclidean lattice. The bounds, analogous to Aronsen’s bounds for uniformly elliptic divergence form diffusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium me...

Percolation with small clusters on random graphs

We consider the problem of finding an induced subgraph in a random dregular graph such that its components have bounded size as the size of the graph gets arbitrarily large. We show that for any threshold τ , the largest size density of such an induced subgraph with component sizes bounded by τ is at most 2(log d)/d for asymptotically large d. A matching lower bound is known for independent set...

On the Spectrum of Lamplighter Groups and Percolation Clusters

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 (integrat...

Random walks on fractal structures and percolation clusters