Necessary and sufficient conditions for universality limits

TL;DR

This paper establishes precise conditions under which universality limits for orthogonal polynomials occur, linking the behavior of the Christoffel-Darboux kernel to tangent measures of the spectral measure, and characterizes various universality classes.

Contribution

It provides necessary and sufficient conditions for universality limits, including new results on tangent measures and a novel homeomorphism between measures and de Branges spaces.

Findings

Christoffel-Darboux kernel has sine kernel asymptotics under specific measure conditions

Universality classes characterized by tangent measures and de Branges spaces

New homeomorphism extends analysis to power law weights and arbitrary measures

Abstract

We derive necessary and sufficient conditions for universality limits for orthogonal polynomials on the real line and related systems. One of our results is that the Christoffel-Darboux kernel has sine kernel asymptotics at a point $\xi$, with regularly varying scaling, if and only if the orthogonality measure (spectral measure) has a unique tangent measure at $\xi$ and that tangent measure is the Lebesgue measure. This includes all prior results with absolutely continuous or singular measures. Our work is not limited to bulk universality; we show that the Christoffel-Darboux kernel has a regularly varying scaling limit with a nontrivial limit kernel if and only if the orthogonality measure has a unique tangent measure at $\xi$ and that tangent measure is not a point mass. The possible limit kernels correspond to homogeneous de Branges spaces; in particular, this equivalence completely characterizes several prominent universality classes such as hard edge universality, Fisher-Hartwig singularities, and jump discontinuities in the weights. The main part of the proof is the derivation of a new homeomorphism. In order to directly apply to the Christoffel-Darboux kernel, this homeomorphism is between measures and chains of de Branges spaces, not between Weyl functions and Hamiltonians. In order to handle limits with power law weights, this homeomorphism goes beyond the more common setting of Poisson-finite measures, and allows arbitrary power bounded measures.