New examples of ballistic RWRE in the low disorder regime

TL;DR

This paper introduces a new criterion for ballistic behavior in low disorder random walks in random environments, extending previous results and providing new examples in three dimensions that do not meet traditional conditions.

Contribution

It presents a novel criterion for ballisticity in low disorder RWRE, expanding the class of known ballistic models beyond Kalikow's condition.

Findings

New criterion for ballisticity based on local drift and environment variance

Construction of new ballistic RWRE examples in dimension 3

Extension of Sznitman's results to low disorder regimes

Abstract

We give a new criterion for ballistic behavior of random walks in random environments which are low disorder perturbations of the simple symmetric random walk on $\mathbb{Z}^d$, for $d\geq 2$. This extends the results established by Sznitman in 2003 and, in particular, allow us to give new examples of ballistic RWREs in dimension $d=3$ which do not satisfy Kalikow's condition. Essentially, this new criterion states that ballisticity occurs whenever the average local drift of the walk is not too small when compared to the standard deviation of the environment. Its proof relies on applying coarse-graining methods together with a variation of the Azuma-Hoeffding concentration inequality in order to verify the fulfillment of a ballisticity condition by Berger, Drewitz and Ram\'irez.