An approach to universality using Weyl m-functions

• 成果类型: Article
• 署名作者: Eichinger, Benjamin; Lukic, Milivoje; Simanek, Brian
• 署名单位: Lancaster University; Technische Universitat Wien; Rice University; Baylor University
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X; 1939-8980
• DOI: 10.4007/annals.2026.203.2.2
• 发表日期: 2026-03
• 页码: 471-510
• 关键词: orthogonal polynomials; universality limits; Christoffel-Darboux kernel; canonical systems; Orthogonal polynomials; bulk universality; christoffel functions; LIMITS; asymptotics; matrices; SUBORDINACY; Respect; kernel; holds

摘要

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.