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On The Cutoff Phenomenon For Dyson-Jacobi Processes

Samuel Chan-Ashing

TL;DR

The paper studies the cutoff phenomenon for the Dyson--Jacobi process on $(0,1)^n$, proving a sharp cutoff in the intrinsic Wasserstein distance with an explicit mixing time $c_n$. The authors introduce a diffeomorphic deformation to flatten the diffusion coefficient, transforming the DJ dynamics into a Euclidean Langevin diffusion while preserving spectral properties and curvature-dimension bounds, thereby enabling sharp mixing-time estimates. They establish a detailed spectral analysis with Jacobi-polynomial eigenfunctions, derive a CD condition in the Euclidean setting, and transfer it back to obtain both lower and upper bounds for convergence, including corollaries across regimes of the coefficient ratio $a_n/b_n$ and connections to other distances. The results connect to beta-Jacobi ensembles, matrix Jacobi processes, and broader beta-ensemble limits (OU/Laguerre), providing sharp, regime-dependent mixing times with implications for random matrix theory and interacting particle systems.

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.

On The Cutoff Phenomenon For Dyson-Jacobi Processes

TL;DR

The paper studies the cutoff phenomenon for the Dyson--Jacobi process on , proving a sharp cutoff in the intrinsic Wasserstein distance with an explicit mixing time . The authors introduce a diffeomorphic deformation to flatten the diffusion coefficient, transforming the DJ dynamics into a Euclidean Langevin diffusion while preserving spectral properties and curvature-dimension bounds, thereby enabling sharp mixing-time estimates. They establish a detailed spectral analysis with Jacobi-polynomial eigenfunctions, derive a CD condition in the Euclidean setting, and transfer it back to obtain both lower and upper bounds for convergence, including corollaries across regimes of the coefficient ratio and connections to other distances. The results connect to beta-Jacobi ensembles, matrix Jacobi processes, and broader beta-ensemble limits (OU/Laguerre), providing sharp, regime-dependent mixing times with implications for random matrix theory and interacting particle systems.

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.
Paper Structure (29 sections, 5 theorems, 67 equations)

This paper contains 29 sections, 5 theorems, 67 equations.

Table of Contents

  1. Introduction and main results
  2. The Dyson--Jacobi process
  3. Existence and uniqueness
  4. Notations
  5. Invariant measure
  6. The intrinsic Wasserstein distance
  7. Main results
  8. Cutoff for other distances or divergences
  9. Matrix cases
  10. Links with the other beta-ensembles
  11. Organization of the paper
  12. Discussions with related works about the cutoff
  13. Cutoff $\mathrm{L}^p$ with $p>1$
  14. Cutoff for TV (for $p=1$)
  15. Diffeomorphic deformations of diffusions
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $x_0$ be in $\mathrm{D}$. Let ${(X^n_t)}_{t\,\geqslant\,0}$ be the Dyson--Jacobi process eq:DJ started at $x_0^n$. Then a cutoff phenomenon occurs for $\hbox{\rm Wasserstein}$ at critical time $c_n$, namely for all $\varepsilon \in (0,1)$, assuming that the following lower bound tends to infinity as $n\to\infty$,

Theorems & Definitions (7)

  • Theorem 1.1: Universal cutoff
  • Corollary 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 3.1: Curvature-dimension