An approach to universality using Weyl m-functions

Eichinger, Benjamin and Lukic, Milivoje and Simanek, Brian (2025) An approach to universality using Weyl m-functions. Annals of Mathematics. ISSN 0003-486X (In Press)

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Abstract

We describe an 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 2×2 transfer matrices such as continuum Schrödinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.