Random walks on supercritical percolation clusters

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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 means that the bound for qₜ(x,·) holds only for t≥ Sₓ(ω), where the constant Sₓ(ω) depends on the percolation configuration ω.