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

TL;DR

This paper establishes a quenched large deviation principle for simple random walks on supercritical percolation clusters, including models with long-range correlations, by developing a unifying approach that handles non-ellipticity and lack of translation invariance.

Contribution

It introduces a novel method for proving quenched LDPs for SRW on percolation models with long-range correlations, extending previous results to non-elliptic, non-translation-invariant settings.

Findings

Proved quenched LDP for SRW on supercritical percolation clusters.

Derived explicit variational formulas for rate functions.

Unified approach applicable to models with long-range correlations.

Abstract

We prove a {\it{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, 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 {\it{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 {\it{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.