Electronic Journal of Probability

Parabolic Harnack inequality and local limit theorem for percolation clusters

Ben Hambly and Martin Barlow

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We consider the random walk on supercritical percolation clusters in $\mathbb{Z}^d$. Previous papers have obtained Gaussian heat kernel bounds, and a.s. invariance principles for this process. We show how this information leads to a parabolic Harnack inequality, a local limit theorem and estimates on the Green's function.

Article information

Electron. J. Probab., Volume 14 (2009), paper no. 1, 1-26.

Accepted: 7 January 2009
First available in Project Euclid: 1 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G50: Sums of independent random variables; random walks
Secondary: 31B05: Harmonic, subharmonic, superharmonic functions

Percolation random walk Harnack inequality local limit theorem

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Hambly, Ben; Barlow, Martin. Parabolic Harnack inequality and local limit theorem for percolation clusters. Electron. J. Probab. 14 (2009), paper no. 1, 1--26. doi:10.1214/EJP.v14-587. https://projecteuclid.org/euclid.ejp/1464819462

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