Random walks on the braid group B_3 and magnetic translations in hyperbolic geometry

TL;DR

This paper investigates random walks on the braid group B_3, analyzing their topological complexity and trivial entanglement probability, by connecting these to magnetic random walks on hyperbolic graphs and quantum translations in hyperbolic geometry.

Contribution

It introduces a novel approach linking braid group random walks with magnetic translations in hyperbolic geometry, providing new insights into their topological and probabilistic properties.

Findings

Computed the drift of random walks on B_3.

Determined the probability of trivial entanglement.

Established a representation of B_3 as magnetic translation operators.

Abstract

We study random walks on the three-strand braid group $B_3$, and in particular compute the drift, or average topological complexity of a random braid, as well as the probability of trivial entanglement. These results involve the study of magnetic random walks on hyperbolic graphs (hyperbolic Harper-Hofstadter problem), what enables to build a faithful representation of $B_3$ as generalized magnetic translation operators for the problem of a quantum particle on the hyperbolic plane.