Asymptotic direction for random walks in mixing random environments

TL;DR

This paper proves that random walks in certain mixing environments with specific conditions have a well-defined asymptotic direction, advancing understanding of their long-term behavior.

Contribution

It establishes the existence of asymptotic directions for random walks under new mixing and polynomial ballisticity conditions.

Findings

Random walks in mixing environments have asymptotic directions.

Conditions include uniform ellipticity, cone mixing, and polynomial ballisticity.

Results apply to a broad class of random environments.

Abstract

We prove that every random walk in a uniformly elliptic random environment satisfying the cone mixing condition and a non-effective polynomial ballisticity condition with high enough degree has an asymptotic direction.