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Brownian motion on reflection quantum groups. Construction and cutoff

Jean Delhaye

TL;DR

This work constructs a Brownian motion on the quantum reflection groups $H_N^{s+}$ by pulling back the Brownian motion on the symmetric quantum group $S_N^+$ through a quotient map and a conditional expectation, yielding a central Lévy process. A key technical step is a reduction to a commutative subalgebra generated by $x_1$, with $x_n = Q_n^s(x_1)$ given by Chebyshev polynomials, whose spectral measure is the free Meixner law $\nu_s$; this enables an explicit cutoff analysis. At time $t_N = N\ln N + cN$, the process restricted to the commutative core converges in moments to a family of measures $\eta_c^s$, and the total-variation distance to Haar tends to $d_{TV}(\eta_c^s,\nu_s)$ for $c>0$, revealing a genuine $s$-dependent cutoff profile. The paper also discusses the case $H_N^{\infty+}$, formulating conjectures about absolute continuity and giving a strategy to obtain lower bounds on the cutoff profile through a subalgebra that captures the same asymptotics. Overall, the results provide a general framework for Brownian motion on quantum reflection groups and illuminate how the parameter $s$ shapes convergence to equilibrium in this noncommutative setting.

Abstract

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

Brownian motion on reflection quantum groups. Construction and cutoff

TL;DR

This work constructs a Brownian motion on the quantum reflection groups by pulling back the Brownian motion on the symmetric quantum group through a quotient map and a conditional expectation, yielding a central Lévy process. A key technical step is a reduction to a commutative subalgebra generated by , with given by Chebyshev polynomials, whose spectral measure is the free Meixner law ; this enables an explicit cutoff analysis. At time , the process restricted to the commutative core converges in moments to a family of measures , and the total-variation distance to Haar tends to for , revealing a genuine -dependent cutoff profile. The paper also discusses the case , formulating conjectures about absolute continuity and giving a strategy to obtain lower bounds on the cutoff profile through a subalgebra that captures the same asymptotics. Overall, the results provide a general framework for Brownian motion on quantum reflection groups and illuminate how the parameter shapes convergence to equilibrium in this noncommutative setting.

Abstract

T. In this study, we construct an analog of the Brownian motion on free reflection quantum groups and compute its cutoff profile.
Paper Structure (9 sections, 11 theorems, 85 equations, 1 figure, 1 table)

This paper contains 9 sections, 11 theorems, 85 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Free reflection quantum groups
  4. The cutoff phenomenon
  5. Defining the Brownian motion on H_N^s+
  6. Computing the limit profile
  7. Finding a commutative subalgebra
  8. The limit profile
  9. The case of H_N^∞+

Key Result

Theorem 1.1

The Brownian motion on $H_N^{s+}$ has cutoff at time $t_N = N\ln N$. More precisely, it exhibits the following cutoff profile: and where $\nu_{s}$ and $\eta_c^s$ are the following probability measures: and

Figures (1)

  • Figure 1: Graph comparing the functions $f_1$, $f_2$, $f_4$, $f_9$, and $f_\infty$ (shown in blue, red, orange, green and purple, respectively).

Theorems & Definitions (31)

  • Theorem 1.1
  • Definition 2.1
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Lemma 2.8
  • proof
  • ...and 21 more