Parabolic Harnack Inequality and Local Limit Theorem for Percolation Clusters

2008/10/14 by Barlow, Martin, Hambly, Ben · 1 citation
#31C25 #60G50 #Analysis of PDEs (math.AP) #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0810.2467

Abstract

We consider the random walk on supercritical percolation clusters in the d-dimensional Euclidean lattice. 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.

Cited by

• Scaling limit of the discrete Gaussian free field with degenerate random conductances

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• Local limit theorems for a directed random walk on the backbone of a supercritical oriented percolation cluster
• Heat Kernel Upper Bounds on Long Range Percolation Clusters
• Simple Random Walk on Long Range Percolation Clusters II: Scaling Limits