Continuity result for the rate function of the simple random walk on supercritical percolation clusters

TL;DR

This paper investigates the continuity of the rate function for the large deviation principle of a simple random walk on supercritical percolation clusters, linking it to the Lyapunov exponent's continuity in the configuration law.

Contribution

It establishes the continuity of the rate function in the law of percolation configurations by analyzing the Lyapunov exponent's properties.

Findings

Lyapunov exponent is continuous in the law of configurations

Rate function continuity is derived from Lyapunov exponent continuity

Provides a link between large deviations and percolation configuration laws

Abstract

We consider the simple random walk on supercritical percolation clusters in the multidimensional cubic lattice. In this model, a quenched large deviation principle holds for the position of the random walk. Its rate function depends on the law of the percolation configuration, and the aim of this paper is to study the continuity of the rate function in the law. To do this, it is useful that the rate function is expressed by the so-called Lyapunov exponent, which is the asymptotic cost paid by the random walk for traveling in a landscape of percolation configurations. In this context, we first observe the continuity of the Lyapunov exponent in the law of the percolation configuration, and then lift it to the rate function.