Random walks veering left

TL;DR

This paper investigates coupled planar random walks with a fixed angular bias, analyzing how the deterministic angle heta influences their behavior and relating the findings to spectral measures of certain random matrices.

Contribution

It introduces a model of coupled random walks with a deterministic angle and computes the Hausdorff dimension of angles leading to atypical walk behaviors.

Findings

Determined the Hausdorff dimension of heta for unusual walk behaviors

Linked the walk model to spectral measures of random matrices

Identified conditions under which the walks exhibit atypical properties

Abstract

We study coupled random walks in the plane such that, at each step, the walks change direction by a uniform random angle plus an extra deterministic angle \theta. We compute the Hausdorff dimension of the \theta for which the walk has an unusual behavior. This model is related to a study of the spectral measure of some random matrices.