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 ${\mathbb{Z}^d}$ (${d \geq 2}$). 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 the vacant set of random interlacements (for ${d \geq 3}$) and the level sets of the Gaussian free field (${d\geq 3}$). Inspired by the methods developed by Kosygina et al. (Commun Pure Appl Math 59:1489–1521, 2006) for proving quenched LDP for elliptic diffusions with a random drift, and by Yilmaz (Commun Pure Appl Math 62(8):1033–1075, 2009) and Rosenbluth (Quenched large deviations for multidimensional random walks in a random environment: a variational formula. Ph.D. thesis, NYU, arXiv:0804.1444v1) 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 of 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 difficulty in our set up lies in the inherent non-ellipticity as well as the lack of translation-invariance stemming from conditioning on the fact that the origin belongs to the infinite cluster. We develop a unifying approach for proving quenched large deviations for SRWPC based on exploiting coercivity properties of the relative entropies in the context of convex variational analysis, combined with input from ergodic theory and invoking geometric properties of the supercritical percolation cluster.