Large deviations for the local times of a random walk among random conductances

TL;DR

This paper establishes a large deviation principle for the local times of a continuous-time random walk in a random conductance environment, accounting for small conductance values and their impact on the walk's behavior.

Contribution

It provides an explicit large deviation principle for local times in a random conductance model with small conductances, linking the rate function to conductance tail behavior.

Findings

Derived an annealed large deviation principle for local times.

Explicitly characterized the rate function in terms of conductance distribution tails.

Identified asymptotics of the principal eigenvalue's lower tails.

Abstract

We derive an annealed large deviation principle for the normalised local times of a continuous-time random walk among random conductances in a finite domain in $\Z^d$ in the spirit of Donsker-Varadhan \cite{DV75}. We work in the interesting case that the conductances may assume arbitrarily small values. Thus, the underlying picture of the principle is a joint strategy of small values of the conductances and large holding times of the walk. The speed and the rate function of our principle are explicit in terms of the lower tails of the conductance distribution. As an application, we identify the logarithmic asymptotics of the lower tails of the principal eigenvalue of the randomly perturbed negative Laplace operator in the domain.