Analysis on Graphs and Fractals

Local limit theorem for random walk on percolation clusters

B Hambly, University of Oxford, UK

Abstract

The random walk on the clusters of supercritical bond percolation is a generic model for diffusion in random media. Heat kernel estimates for the random walk are known and there is also an almost sure invariance principle for the convergence of the walks to Brownian motion under rescaling.  We prove a prabolic Harnack inequality and use it to obtain a local limit theorem for the convergence under rescaling of the heat kernel of the random walks to the Gaussian heat kernel.