Two-dimensional anisotropic random walks: fixed versus random column configurations for transport phenomena

TL;DR

This paper studies two types of two-dimensional anisotropic random walks on a square lattice, establishing invariance principles that connect these walks to anisotropic Brownian motions, with a focus on fixed versus random column configurations.

Contribution

It provides a unified analysis of fixed and random column configurations for anisotropic random walks, proving invariance principles with detailed proofs and historical context.

Findings

Weak Donsker invariance principle established for fixed and random configurations.

Strong Strassen invariance principle demonstrated for both walk types.

Connections made between random walks and anisotropic Brownian motions.

Abstract

We consider random walks on the square lattice of the plane along the lines of Heyde (1982, 1993) and den Hollander (1994), whose studies have in part been inspired by the so-called transport phenomena of statistical physics. Two-dimensional anisotropic random walks with anisotropic density conditions a' la Heyde (1982, 1993) yield fixed column configurations and nearest-neighbour random walks in a random environment on the square lattice of the plane as in den Hollander (1994) result in random column configurations. In both cases we conclude simultaneous weak Donsker and strong Strassen type invariance principles in terms of appropriately constructed anisotropic Brownian motions on the plane, with self-contained proofs in both cases. The style of presentation throughout will be that of a semi-expository survey of related results in a historical context.