A local limit theorem in stationary random environment of conductances on Z

TL;DR

This paper establishes a local limit theorem for nearest neighbor random walks in a stationary random conductance environment on Z, without relying on uniform ellipticity or independence assumptions, using discrete Nash-type inequalities.

Contribution

It introduces a novel approach to proving local limit theorems without classic assumptions, expanding understanding of random walks in complex environments.

Findings

Proves a local limit theorem in stationary random conductance environments.

Extends results beyond uniform ellipticity and independence assumptions.

Utilizes discrete Nash-type inequalities for the proof.

Abstract

We prove a local limit theorem for nearest neighbours random walks in stationary random environment of conductances on Z without using any of both classic assumptions of uniform ellipticity and independence on the conductances. Besides the central limit theorem, we use discrete differential "Nash-type inequalities" associated with the Hausdorff's representation of the completely decreasing sequences.