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Brownian Motion on the Unitary Quantum Group: Construction and Cutoff

Jean Delhaye

TL;DR

This work extends the cutoff phenomenon to the unitary free quantum group $U_N^+$ by constructing a Brownian-type process via centralized Gaussian generating functionals and analyzing its asymptotic mixing. A key innovation is restricting attention to a commutative subalgebra generated by a single combining observable, which yields tractable moment convergence to a family of mixed laws $\eta_c^r$ that blend the semicircle law with a parametric singular component. The main result identifies the limit profile as the total-variation distance between $\eta_c^r$ and $\nu_{SC}$, with absolute continuity occurring for $c>0$ and a nontrivial singular part appearing for $c\le 0$, and it shows the profile depends injectively on the parameter $r$ (including the special cases $r=0$ and $r=\infty$). The paper also highlights a no-cutoff instance when the process is driven purely by the $\beta$-term, illustrating the nuanced behavior of quantum group diffusion versus classical groups. Altogether, these findings extend previously known cutoff results for $O_N^+$ and $S_N^+$ to the unitary quantum setting and deepen understanding of quantum probabilistic diffusion on noncommutative spaces.

Abstract

In this study, we construct an analog of the Brownian motion on free unitary quantum groups and compute its cutoff profile.

Brownian Motion on the Unitary Quantum Group: Construction and Cutoff

TL;DR

This work extends the cutoff phenomenon to the unitary free quantum group by constructing a Brownian-type process via centralized Gaussian generating functionals and analyzing its asymptotic mixing. A key innovation is restricting attention to a commutative subalgebra generated by a single combining observable, which yields tractable moment convergence to a family of mixed laws that blend the semicircle law with a parametric singular component. The main result identifies the limit profile as the total-variation distance between and , with absolute continuity occurring for and a nontrivial singular part appearing for , and it shows the profile depends injectively on the parameter (including the special cases and ). The paper also highlights a no-cutoff instance when the process is driven purely by the -term, illustrating the nuanced behavior of quantum group diffusion versus classical groups. Altogether, these findings extend previously known cutoff results for and to the unitary quantum setting and deepen understanding of quantum probabilistic diffusion on noncommutative spaces.

Abstract

In this study, we construct an analog of the Brownian motion on free unitary quantum groups and compute its cutoff profile.
Paper Structure (13 sections, 24 theorems, 167 equations, 1 figure)

This paper contains 13 sections, 24 theorems, 167 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Free unitary quantum groups
  4. The cutoff phenomenon
  5. The Quantum Unitary Brownian Motion
  6. Defining the quantum unitary Brownian motion
  7. Computing the values of the Brownian motion
  8. Restricting the Study to a Smaller Algebra
  9. Moment Convergence
  10. A Special Case Without Cutoff
  11. The Limit Profile
  12. Absolute continuity
  13. Computation of the limit profile

Key Result

Theorem 2.2

There is a unique state $h$ on $U_N^+$ such that It is called the Haar state of $U_N^+$.

Figures (1)

  • Figure 1: Total variation distance $g_r$ for increasing $r$ (top to bottom).

Theorems & Definitions (55)

  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Remark 3.1
  • ...and 45 more