Cut-off phenomenon for random walks on free orthogonal quantum groups

TL;DR

This paper establishes the cut-off phenomenon for certain random walks on free orthogonal quantum groups, providing bounds on convergence and identifying the cutoff point, marking a first for genuine compact quantum groups.

Contribution

It introduces the first analysis of the cut-off phenomenon for genuine compact quantum groups, specifically for free orthogonal quantum groups and related models.

Findings

Identifies the cutoff at N ln(N) / 2(1 - cos(θ)) for the quantum group walk.

Provides bounds in total variation distance for these quantum walks.

Extends results to mixtures of rotations and quantum permutations.

Abstract

We give bounds in total variation distance for random walks associated to pure central states on free orthogonal quantum groups. As a consequence, we prove that the analogue of the uniform plane Kac walk on this quantum group has a cut-off at $N\ln(N)/2(1-\cos(\theta))$. This is the first result of this type for genuine compact quantum groups. We also obtain similar results for mixtures of rotations and quantum permutations.