Random walks on supercritical percolation clusters

Abstract

We obtain Gaussian upper and lower bounds on the transition density q_t(x,y) of the continuous time simple random walk on a supercritical percolation cluster C_{\infty} in the Euclidean lattice. The bounds, analogous to Aronsen's bounds for uniformly elliptic divergence form diffusions, hold with constants c_i depending only on p (the percolation probability) and d. The irregular nature of the medium means that the bound for q_t(x,\cdot) holds only for t\ge S_x(\omega), where the constant S_x(\omega) depends on the percolation configuration \omega.Comment: Published at http://dx.doi.org/10.1214/009117904000000748 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org