Elliot Paquette -- The edge scaling limit of the Gaussian beta-ensemble characteristic polynomial
Channel: Matrices Et Graphes Aléatoires GDR CNRS
Date: 2021-01-12

Abstract: The Gaussian beta-ensemble (GbetaE) is a 1-parameter generalization of the Gaussian orthogonal/unitary/symplectic ensembles which retains some integrable structure. Using this ensemble, in Ramirez, Rider and Virag constructed a limiting point process, the Airy-beta point process, which is the weak limit of the point process of eigenvalues in a neighborhood of the spectral edge. They constructed a limiting Sturm—Liouville problem, the stochastic Airy equation with Dirichlet boundary conditions, and they proved convergence of a discrete operator with spectra given by GbetaE to this limit. Jointly with Gaultier Lambert, we give a construction of a new limiting object, the stochastic Airy function (SAi); we also show this is the limit of the characteristic polynomial of GbetaE in a neighborhood of the edge. It is the solution of the stochastic Airy equation, which is the usual Airy equation perturbed by a multiplicative white noise, with specified asymptotics at time=+infinity. Its zeros are given by the Airy-beta point process, and the mode of convergence we establish provides a new proof that Airy-beta is the limiting point process of eigenvalues of GbetaE. In this talk, we survey what new information we have on the characteristic polynomial; we show from where the stochastic Airy equation arises; we show how SAi is constructed; and we leave some unanswered questions.