Quenched large deviations for simple random walks on percolation clusters including long-range correlations

Abstract

We prove a quenched large deviation principle (LDP)for a simple random walk on a supercritical percolation cluster (SRWPC) on the lattice.The models under interest include classical Bernoulli bond and site percolation as well as models that exhibit long range correlations, like the random cluster model, the random interlacement and its vacant set and the level sets of the Gaussian free field. Inspired by the methods developed by Kosygina, Rezakhanlou and Varadhan ([KRV06]) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz ([Y08]) and Rosenbluth ([R06]) for similar results regarding elliptic random walks in random environment, we take the point of view of the moving particle and prove a large deviation principle for the quenched distribution of the pair empirical measures if the environment Markov chain in the non-elliptic case of SRWPC. Via a contraction principle, this reduces easily to a quenched LDP for the distribution of the mean velocity of the random walk and both rate functions admit explicit variational formulas. The main approach of our proofs are based on exploiting coercivity properties of the relative entropy in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the percolation cluster under supercriticality.