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, 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 or a random matrix in a neighborhood of the spectral edge.
Jointly with Gaultier Lambert, we give a construction of a new limiting object, the stochastic Airy function (SAi); we show this is the limit of the characteristic polynomial of GbetaE in a neighborhood of the spectral edge. It is the bounded solution of the stochastic Airy equation, which is the usual Airy equation perturbed by a multiplicative white noise.