The Tracy-Widom $\beta = 2$ distribution is the marginal distribution of the top curve in the Airy line ensemble, which consists of an infinite sequence of random curves introduced by Prähofer and Spohn. This ensemble was conjectured to describe the scaling limit of various random surfaces and stochastic growth models within the Kardar–Parisi–Zhang (KPZ) universality class. More generally, the Tracy-Widom $\beta$ distribution represents the marginal distribution of the top curve in the Airy $\beta$ line ensemble, which arises from the scaling limit of Dyson's Brownian motion.

In this talk, we will present a characterization result for the Airy $\beta$ line ensemble via a stochastic differential equation and discuss its applications, particularly in the convergence of the Airy $\beta$ line ensemble for Dyson Brownian motions with general potentials, Laguerre processes, and Jacobi processes. These results are based on joint work with Lingfu Zhang.