# Quantum reflections, random walks and cut-off

## TL;DR

This paper investigates the cut-off phenomenon in quantum random walks on free unitary quantum groups, extending classical results to quantum settings and exploring related quantum reflection groups.

## Contribution

It introduces a quantum analogue of classical cut-off results for random walks on quantum groups and studies their behavior on quantum reflection groups and free wreath products.

## Key findings

- Established a quantum cut-off phenomenon for certain quantum random walks.
- Extended classical reflection group results to quantum reflection groups.
- Analyzed random walks on free wreath products of finite groups and quantum permutation groups.

## Abstract

We study the cut-off phenomenon for random walks on free unitary quantum groups coming from quantum conjugacy classes of classical reflections. We obtain in particular a quantum analogue of the result of U. Porod concerning certain mixtures of such reflections. We also study random walks on quantum reflection groups and more generally free wreath products of finite group by quantum permutation groups.