Applications of optimal transport to Dyson Brownian Motions and beyond

TL;DR

This paper introduces a novel optimal transport-based approach to derive sharp continuity estimates for various random curve ensembles, including Dyson Brownian motions and related models, extending the applicability of these techniques.

Contribution

It develops a new method using Caffarelli's contraction theorem to obtain uniform modulus of continuity estimates for a broad class of log-concave perturbations of Brownian motions.

Findings

Sharp continuity estimates for $eta$-Dyson Brownian motions with $eta \, \geq \, 2$

Extension of methods to Airy, KPZ, and O'Connell-Yor line ensembles

Unified approach for log-concave perturbations of Brownian motions

Abstract

We develop a new method based on Caffarelli's contraction theorem in optimal transport to obtain sharp and uniform modulus of continuity estimates for $\beta$-Dyson Brownian motions with $\beta \geq 2$. Our method extends to a large class of random curve collections, which can be viewed as log-concave perturbations of Brownian motions, including the $\beta$-Dyson Brownian motion, the Air$\text{y}_{\beta}$ line ensemble, the KPZ line ensemble, and the O'Connell-Yor line ensemble.