Title: Universality limits for orthogonal polynomials
Abstract: It is often expected that the local statistical behavior of eigenvalues of some system depends only on its local properties; for instance, the local distribution of zeros of orthogonal polynomials should depend only on the local properties of the measure of orthogonality. This phenomenon is studied using an object called the Christoffel-Darboux kernel. The most commonly studied case is known as bulk universality, where the rescaled limit of Christoffel-Darboux kernels converges to the sine kernel. In this talk, we will survey this subject, prior results, and a recent result which gives for the first time a completely local sufficient condition for bulk universality. The new approach is based on a matrix version of the Christoffel-Darboux kernel and the de Branges theory of canonical systems, and it applies to other self-adjoint systems with 2×2 transfer matrices such as continuum Schrodinger and Dirac operators. The talk is based on joint work with Benjamin Eichinger (Technical University Wien) and Brian Simanek (Baylor University).
The Dalhousie-AARMS Analysis-Applied Math-Physics Seminar takes place on Fridays from 4 – 5 pm Atlantic Time over either Zoom and/or in Chase 227 depending on the speaker. If you would like to attend, please email the organizers for connection details.