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Random Walks on Supercritical Percolation Clusters

Martin T. Barlow
The Annals of Probability
Vol. 32, No. 4 (Oct., 2004), pp. 3024-3084
Stable URL: http://www.jstor.org/stable/3481514
Page Count: 61
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
Random Walks on Supercritical Percolation Clusters
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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 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.

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