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 cidepending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for qt(x,·) holds only for t≥ Sx(ω), where the constant Sx(ω) depends on the percolation configuration ω.