Mathematics

The Gaussian β-ensemble is one of the central model in random matrix theory. Because of its integrable structure, it allows to describe several universal limiting laws of the eigenvalues of random matrices. For instance, in a seminal work, Ramirez-Rider-Virag constructed the Airy-β process, the scaling limit of the eigenvalues near the spectral edge of the Gaussian β-ensemble and gave a new representation for the Tracy-Widom distributions.In this talk, I intend to review this construction and present recent results on the asymptotics for the characteristic polynomial of the Gaussian β-ensemble obtained jointly with Elliot Paquette (McGill University). Our results rely on a new approach to study the characteristic polynomial based on its recurrence.