# Quenched invariance principle for random walks among random degenerate conductances

## TL;DR

This paper proves a quenched invariance principle for random walks in a stationary ergodic environment with random conductances, under minimal moment conditions, extending previous results and ensuring sublinear correctors.

## Contribution

It establishes the quenched invariance principle under optimal moment conditions, improving earlier work and demonstrating sublinearity of the corrector everywhere.

## Key findings

- Proves quenched invariance principle under minimal moment conditions.
- Shows the corrector is sublinear everywhere.
- Provides a deterministic local boundedness result for divergence form equations.

## Abstract

We consider the random conductance model in a stationary and ergodic environment. Under suitable moment conditions on the conductances and their inverse, we prove a quenched invariance principle for the random walk among the random conductances. The moment conditions improve earlier results of Andres, Deuschel and Slowik [Ann.\ Probab.] and are the minimal requirement to ensure that the corrector is sublinear everywhere. The key ingredient is an essentially optimal deterministic local boundedness result for finite difference equations in divergence form.