Parabolic harnack inequality and local limit theorem for percolation clusters

Abstract

We consider the random walk on supercritical percolation clusters in ℤd . 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. © 2009 Applied Probability Trust.