Homogenization and quenched large deviations

We prove a quenched large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster on Z^d, d ≥ 2. We take the point of view of the moving particle and first prove a quenched LDP for the distribution of the pair empirical measures of the environment Markov chain. The classical LDP for the distribution of the mean velocity of the random walk drops out easily via the contraction principle and both rate functions admit variational formulas.

Our results are based on invoking ergodicity arguments in this non-elliptic set up to control the growth of gradient functions (correctors) which come up naturally via convex variational analysis in the context of homogenization of random Hamilton-Jacobi Bellman equations.

This is joint work with Noam Berger (Munich).