Abstract
In this talk, we focus on an infinite-dimensional model of repulsively interacting Brownian motions: Dyson Brownian motion (DBM) at soft-edge scaling. It is known that its stationary process is the Airy line ensemble, a collection of non-intersecting random curves linked to many models in the KPZ universality class. We show that its time-marginal law is characterised as a Wasserstein steepest gradient descent of the relative entropy in the space of probability measures over the configuration space — an infinite-dimensional analogue of Jordan-Kinderlehrer-Otto/Ambrosio-Gigli-Savaré theory. From a metric-geometric viewpoint, our result shows that the configuration space endowed with invariant measure of the DBM (i.e., Airy_2 point process) is an RCD space, a space having a uniform Ricci curvature lower bound in the sense of Lott-Villani/Sturm and Gigli. As an application, we establish various new functional inequalities (e.g., HWI, distorted Brunn-Minkowski, dimension-free Harnack) for the model. Furthermore, we discover the new phenomenon that the time-marginal law exhibits number rigidity in the sense of Ghosh and Peres (i.e., the number of particles inside a box is determined by the configuration outside), revealing a formation of a random crystal-like structure by the DBM. This talk is based on arXiv:2509.06869.
