Quenched Large Deviations for Simple Random Walks on Supercritical Percolation Clusters

TL;DR

This paper establishes a quenched large deviation principle for simple random walks on supercritical percolation clusters in
76^d, providing explicit formulas for rate functions and employing ergodic and homogenization techniques.

Contribution

It introduces a novel quenched LDP for random walks on percolation clusters, utilizing gradient functions distinct from classical correctors, with explicit variational rate formulas.

Findings

Proves a quenched LDP for the walk's pair empirical measures.

Derives a quenched LDP for the mean velocity distribution.

Provides explicit variational formulas for rate functions.

Abstract

We prove a {\it{quenched}} large deviation principle (LDP) for a simple random walk on a supercritical percolation cluster on $\Z^d$, $d\geq 2$.. We take the point of view of the moving particle and first prove a quenched LDP for the distribution of the {\it{pair empirical measures}} of the environment Markov chain. 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. Our results are based on invoking ergodicity arguments in this non-elliptic set up to control the growth of {\it{gradient functions (correctors)}} which come up naturally via convex variational analysis in the context of homogenization of random Hamilton Jacobi Bellman equations along the arguments of Kosygina, Rezakhanlou and Varadhan (\cite{KRV06}). Although enjoying some similarities, our gradient function is structurally different from {\it{the}} classical {\it{Kipnis-Varadhan corrector}}, a well-studied object in the context of reversible random motions in random media.