Applications of optimal transport to Dyson Brownian Motions and beyond
Xuan Wu
TL;DR
This work develops a new, sharp approach to modulus-of-continuity estimates for $\beta$-Dyson Brownian motions with $\beta\ge 2$ by leveraging Caffarelli's contraction theorem from optimal transport. By viewing these processes as log-concave perturbations of Brownian motion, the authors derive direct Gaussian-type comparisons that yield explicit, uniform bounds across all layers and extend the method to a broad class $\mathbf{LC}$ of log-concave-perturbed curves. The LC framework encompasses the Airy$_{\beta}$ line ensemble, the O'Connell-Yor line ensemble, and the KPZ line ensemble, establishing both finite-dimensional log-concavity and modulus-of-continuity results that support tightness and scaling limits. These results have significant implications for the construction of the directed landscape and for KPZ universality, providing rigorous control that connects random matrix theory, Brownian/Lévy-type line ensembles, and stochastic growth models.
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 $β$-Dyson Brownian motions with $β\geq 2$. Our method extends to a large class of random curve collections, which can be viewed as log-concave perturbations of Brownian motions, including the $β$-Dyson Brownian motion, the Air$\text{y}_β$ line ensemble, the KPZ line ensemble, and the O'Connell-Yor line ensemble.