The stochastic Airy operator and an interesting eigenvalue process

Abstract: The Gaussian ensembles, originally introduced by Wigner may be generalized to an n-point ensemble called the beta-Hermite ensemble. As with the original ensembles we are interested in studying the local behavior of the eigenvalues. At the edges of the ensemble the rescaled eigenvalues converge to the Airy_beta process which for general beta is characterized as the eigenvalues of a certain random differential operator called the stochastic Airy operator (SAO). In this talk I will give a short introduction to the Stochastic Airy Operator and the proof of convergence of the eigenvalues, before introducing another interesting eigenvalue process. This process can be characterized as a limit of eigenvalues of minors of the tridiagonal matrix model associated to the beta-Hermite ensemble as well as the process formed by the eigenvalues of the SAO under a restriction of the domain. This is joint work with Angelica Gonzalez.

mathematical physicsprobability

Audience: researchers in the discipline

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