A local limit theorem for random walks in balanced environments

TL;DR

This paper establishes a local limit theorem for one-dimensional recurrent balanced random walks in random environments, extending the understanding of local behaviors beyond the previously studied ballistic case.

Contribution

It proves a local limit theorem for recurrent balanced environments, filling a gap in the theory of random walks in random environments.

Findings

Proves a local limit theorem for recurrent balanced environments.

Completes the understanding of local limits in one-dimensional random walks.

Uses a novel proof method different from the ballistic case.

Abstract

Central limit theorems for random walks in quenched random environments have attracted plenty of attention in the past years. More recently still, finer local limit theorems -- yielding a Gaussian density multiplied by a highly oscillatory modulating factor -- for such models have been obtained. In the one-dimensional nearest-neighbor case with i.i.d. transition probabilities, local limits of uniformly elliptic ballistic walks are now well understood. We complete the picture by proving a similar result for the only recurrent case, namely the balanced one, in which such a walk is diffusive. The method of proof is, out of necessity, entirely different from the ballistic case.