Universal cutoff for Dyson Ornstein Uhlenbeck process

TL;DR

This paper investigates the convergence to equilibrium of the Dyson-Ornstein-Uhlenbeck process, revealing a cutoff phenomenon at a critical time independent of interaction strength, with implications for high-dimensional heat kernel analysis.

Contribution

It demonstrates a universal cutoff time for the Dyson-Ornstein-Uhlenbeck process, independent of the interaction parameter, and provides a detailed analysis of the non-interacting case.

Findings

Cutoff phenomenon occurs at a universal critical time.

Critical time is independent of the interaction parameter beta.

Complete analysis of the non-interacting Ornstein-Uhlenbeck process.

Abstract

We study the Dyson-Ornstein-Uhlenbeck diffusion process, an evolving gas of interacting particles. Its invariant law is the beta Hermite ensemble of random matrix theory, a non-product log-concave distribution. We explore the convergence to equilibrium of this process for various distances or divergences, including total variation, relative entropy, and transportation cost. When the number of particles is sent to infinity, we show that a cutoff phenomenon occurs: the distance to equilibrium vanishes abruptly at a critical time. A remarkable feature is that this critical time is independent of the parameter beta that controls the strength of the interaction, in particular the result is identical in the non-interacting case, which is nothing but the Ornstein-Uhlenbeck process. We also provide a complete analysis of the non-interacting case that reveals some new phenomena. Our work relies among other ingredients on convexity and functional inequalities, exact solvability, exact Gaussian formulas, coupling arguments, stochastic calculus, variational formulas and contraction properties. This work leads, beyond the specific process that we study, to questions on the high-dimensional analysis of heat kernels of curved diffusions.