An approach to universality using Weyl m-functions

Event time: 
Thursday, November 18, 2021 - 4:15pm
Milivoje Lukic
Speaker affiliation: 
Rice University
Event description: 


    In this talk, I will present joint work with Benjamin

    Eichinger and Brian Simanek: a new approach to universality limits for

    orthogonal polynomials on the real line which is completely local and

    uses only the boundary behavior of the Weyl m-function at the point.

    We show that bulk universality of the Christoffel–Darboux kernel

    holds for any point where the imaginary part of the m-function has a

    positive finite nontangential limit. This approach is based on

    studying a matrix version of the Christoffel–Darboux kernel and the

    realization that bulk universality for this kernel at a point is

    equivalent to the fact that the corresponding m-function has normal

    limits at the same point. Our approach automatically applies to other

    self-adjoint systems with 2x2 transfer matrices such as continuum

    Schrodinger and Dirac operators. We also obtain analogous results for

    orthogonal polynomials on the unit circle.