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arXiv · 2108.01629

An approach to universality using Weyl m-functions

Benjamin Eichinger·Milivoje Lukić·Brian Simanek
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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 $2\times 2$ transfer matrices such as continuum Schr\"odinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.

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Benjamin Eichinger, Milivoje Lukić, Brian Simanek. 2021-08-03. An approach to universality using Weyl m-functions. https://arxiv.org/abs/2108.01629

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