The Edge Scaling Limit of the Characteristic Polynomial of the Gaussian β-Ensembles
In this talk, I report on the asymptotics of the characteristic polynomials of the Gaussian β-ensembles for general β > 0. Based on the Dumitriu-Edelman matrix models for the Gaussian β-ensemble, I will present a probabilistic coupling between the characteristic polynomial, a Gaussian analytic function and a new object called the stochastic Airy function. This random entire function arise as the scaling limit of the characteristic polynomial at the spectral edge and its zero set is exactly the Airy-β point process. This is joint work with Elliot Paquette and our results are based on the study of the transfer matrix recurrence satisfied by the characteristic polynomials.