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An approach to universality using Weyl m-functions
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DOI：10.4007/annals.2026.203.2.2.png)
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Abstract
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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 Schro & uml;dinger and Dirac operators. We also obtain analogous results for orthogonal polynomials on the unit circle.
Keywords:
orthogonal polynomials
universality limits
Christoffel-Darboux kernel
canonical systems
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