Cutoff profiles for quantum L\'{e}vy processes and quantum random transpositions

TL;DR

This paper investigates cutoff phenomena in quantum Lévy processes on free orthogonal quantum groups and quantum permutations, revealing precise cutoff times and profiles involving free Poisson and semi-circle distributions.

Contribution

It introduces the first analysis of cutoff profiles for quantum Lévy processes, extending classical results to quantum groups and identifying explicit distributional behaviors.

Findings

Cutoff at time N log N for quantum orthogonal groups

Explicit cutoff profile involving free Poisson and semi-circle laws

Results extend classical cutoff phenomena to quantum settings

Abstract

We consider a natural analogue of Brownian motion on free orthogonal quantum groups and prove that it exhibits a cutoff at time $N\ln(N)$. Then, we study the induced classical process on the real line and compute its atoms and density. This enables us to find the cutoff profile, which involves free Poisson distributions and the semi-circle law. We prove similar results for quantum permutations and quantum random transpositions.