The Annals of Probability

Random walks on supercritical percolation clusters

Martin T. Barlow

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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 ${\mathcal{C}}_{\infty}$ 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 tSx(ω), where the constant Sx(ω) depends on the percolation configuration ω.

Article information

Ann. Probab. Volume 32, Number 4 (2004), 3024-3084.

First available in Project Euclid: 8 February 2005

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K37: Processes in random environments
Secondary: 58J35: Heat and other parabolic equation methods

Percolation random walk heat kernel


Barlow, Martin T. Random walks on supercritical percolation clusters. Ann. Probab. 32 (2004), no. 4, 3024--3084. doi:10.1214/009117904000000748.

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