Asymptotic direction of a ballistic random walk in a two-dimensional random environment with nonuniform mixing

TL;DR

This paper proves the existence of an asymptotic direction for a biased random walk in a two-dimensional environment with polynomially decaying correlations, extending previous methods with new ideas.

Contribution

It introduces a novel approach combining renormalization and environment-specific techniques to establish asymptotic direction in correlated 2D environments.

Findings

Existence of asymptotic direction for the walk.

Application to classical models with correlated environments.

Extension of renormalization methods to new framework.

Abstract

In this paper, we study random walks evolving with a directional bias in a two-dimensional random environment with correlations that vanish polynomially. Using renormalization methods first employed for one-dimensional dynamic environments along with additional ideas specific to this new framework, we show that there exists an asymptotic direction for such a random walk. We also provide examples of classical models for which our results apply.