Jiaming Xu: Airy Beta line ensemble through Dunkl operators

Abstract:

For any real number \(\beta>0\), the \(\operatorname{Airy}_{\beta}\) line ensemble is an infinite collection of random continuous curves that serves as a universal edge scaling limit of Beta ensembles. For \(\beta=2\), this object plays a key role in KPZ class and satisfies the Brownian Gibbs property, while for general \(\beta\) much less is known. We construct \(\operatorname{Airy}_{\beta}\) line ensemble as the edge limit of two objects: \(\operatorname{Dyson}_{\beta}\) Brownian motion and \(\operatorname{Gaussian}_{\beta}\) corner process, and give the first explicit expression of its multi-time Laplace transform. In contrast to the previous SDE characterization, our approach relies on the moment method and the actions of Dunkl operators, which originate in symmetric function theory, and are applied to random matrices in recent years. Joint work with Vadim Gorin and Lingfu Zhang.