Parabolic Harnack Inequality and Local Limit Theorem for Percolation   Clusters
  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. 
Martin Barlow and Ben Hambly