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.
Electron. J. Probab.
14:
1-26
(2009).
DOI: 10.1214/EJP.v14-587