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A new approach for the unitary Dyson Brownian motion through the theory of viscosity solutions

Charles Bertucci, Valentin Pesce

TL;DR

The paper addresses the unitary Dyson Brownian motion on the circle and develops a viscosity-solution framework for its primitive Dyson equation on $\\mathbb{T}$. By deriving the primitive equation $\partial_t F+(\partial_\theta F)A_0[F]=0$ and proving a comparison principle, it establishes well-posedness and analyzes long-time behavior. A discrete-periodic particle scheme coupled with Itô calculus proves convergence of the empirical spectral flow to the unique viscosity solution, while monotonicity, $L^\infty$ regularization, and free-entropy properties yield convergence toward the uniform circular measure with explicit rates in various norms. The results provide a rigorous PDE-based understanding of unitary Coulomb gas dynamics on the circle, including equilibrium and regularity properties, with potential implications for spectral theory and random matrix dynamics on compact manifolds.

Abstract

In this paper, we study the unitary Dyson Brownian motion through a partial differential equation approach recently introduced for the real Dyson case. The main difference with the real Dyson case is that the spectrum is now on the circle and not on the real line, which leads to particular attention to comparison principles. First we recall why the system of particles which are the eigenvalues of unitary Dyson Brownian motion is well posed thanks to a containment function. Then we proved that the primitive of the limit spectral measure of the unitary Dyson Brownian motion is the unique solution to a viscosity equation obtained by primitive the Dyson equation on the circle. Finally, we study some properties of solutions of Dyson's equation on the circle. We prove a L $\infty$ regularization. We also look at the long time behaviour in law of a solution through a study of the so-called free entropy of the system. We conclude by discussing the uniform convergence towards the uniform measure on the circle of a solution of the Dyson equation.

A new approach for the unitary Dyson Brownian motion through the theory of viscosity solutions

TL;DR

The paper addresses the unitary Dyson Brownian motion on the circle and develops a viscosity-solution framework for its primitive Dyson equation on . By deriving the primitive equation and proving a comparison principle, it establishes well-posedness and analyzes long-time behavior. A discrete-periodic particle scheme coupled with Itô calculus proves convergence of the empirical spectral flow to the unique viscosity solution, while monotonicity, regularization, and free-entropy properties yield convergence toward the uniform circular measure with explicit rates in various norms. The results provide a rigorous PDE-based understanding of unitary Coulomb gas dynamics on the circle, including equilibrium and regularity properties, with potential implications for spectral theory and random matrix dynamics on compact manifolds.

Abstract

In this paper, we study the unitary Dyson Brownian motion through a partial differential equation approach recently introduced for the real Dyson case. The main difference with the real Dyson case is that the spectrum is now on the circle and not on the real line, which leads to particular attention to comparison principles. First we recall why the system of particles which are the eigenvalues of unitary Dyson Brownian motion is well posed thanks to a containment function. Then we proved that the primitive of the limit spectral measure of the unitary Dyson Brownian motion is the unique solution to a viscosity equation obtained by primitive the Dyson equation on the circle. Finally, we study some properties of solutions of Dyson's equation on the circle. We prove a L regularization. We also look at the long time behaviour in law of a solution through a study of the so-called free entropy of the system. We conclude by discussing the uniform convergence towards the uniform measure on the circle of a solution of the Dyson equation.

Paper Structure

This paper contains 28 sections, 30 theorems, 228 equations.

Table of Contents

  1. Introduction
  2. The Dyson model: from random matrices to particles in interaction
  3. The Dyson model
  4. Notation and well-posedness of the model
  5. Notation
  6. Formulas
  7. Condition for well-posedness of the model
  8. Study of the Dyson partial differential equation on the circle
  9. Notation
  10. Primitive equation
  11. Uniqueness of viscosity solution
  12. Convergence of the system of particles
  13. Periodic system of $N$ particles
  14. Discrete comparison principle
  15. Convergence of the system of particles
  16. ...and 13 more sections

Key Result

Theorem 2.3

For $N\ge 1$, if $\beta_N\ge N^2/2$, then for any initial data $\lambda_0\in \mathbb{R}^N_>$, there exists a unique strong solution to $~dysoncircleeq$ defined on $[0,+\infty)$ and $T_{\partial D}=+\infty$ almost surely.

Theorems & Definitions (66)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3: Criterion for absence of explosion
  • proof
  • Lemma 2.4
  • proof
  • Theorem 2.5: Non explosion of the Dyson model
  • Remark 2.6
  • Remark 3.1
  • Remark 3.2
  • ...and 56 more