Random walks on supercritical percolation clusters

2003/02/28 by Martin T. Barlow · 8 citations
Mathematics · #math.PR #msc:60K37 #msc:58J35.

paper · pdf · doi:10.1214/009117904000000748

published as Annals of Probability 2004, Vol. 32, No. 4, 3024-3084 · 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)

arxiv created 2005/04/06 · arxiv updated 2009/11/30

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_∞ 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,⋅) holds only for t≥ S_x(ω), where the constant S_x(ω) depends on the percolation configuration ω.

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