Random walks on supercritical percolation clusters

Abstract

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 C1 in the Euclidean lattice. The bounds, analogous to Aronsen's bounds for uniformly elliptic divergence form diusions, hold with constants ci depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for qt(x; ) only holds for t Sx(!), where the constant Sx(!) depends on the percolation congura- tion !.