On The Cutoff Phenomenon For Dyson-Jacobi Processes

TL;DR

This paper investigates the rapid convergence to equilibrium of Dyson-Jacobi processes, revealing a cutoff phenomenon and providing explicit mixing times through geometric and analytical techniques.

Contribution

It establishes the occurrence of a cutoff phenomenon for Dyson-Jacobi processes and derives explicit formulas for mixing times using geometric and curvature-based methods.

Findings

Confirmed the cutoff phenomenon for Dyson-Jacobi processes.

Derived explicit formulas for mixing times.

Applied geometric methods to analyze convergence.

Abstract

We study the convergence to equilibrium of the Dyson-Jacobi process, a system of n interacting particles on the segment [0, 1] arising from Random Matrix Theory. We establish the occurence of a cutoff phenomenon for the intrinsic Wasserstein distance and provide an explicit formula for the associated mixing time. Our approach relies on the interplay between the Riemannian geometry of the process and a flattened Euclidean representation obtained via a diffeomorphic deformation. This transformation allows us to transfer curvature-dimension inequalities from the Euclidean setting to the original space, thereby yielding sharp quantitative estimates.