The Nevai condition and a local law of large numbers for orthogonal polynomial ensembles

TL;DR

This paper establishes global and local laws of large numbers for orthogonal polynomial ensembles, using concentration inequalities and the Nevai condition, under weak assumptions on the measure.

Contribution

It introduces a new approach combining concentration inequalities and the Nevai condition to analyze asymptotics of orthogonal polynomial ensembles.

Findings

Proved global and local laws of large numbers for these ensembles.

Extended the Nevai condition to broader settings.

Provided weak conditions on the measure for asymptotic results.

Abstract

We consider asymptotics of orthogonal polynomial ensembles, in the macroscopic and mesoscopic scales. We prove both global and local laws of large numbers (analogous to the recently proven local semicircle law for Wigner matrices) under fairly weak conditions on the underlying measure $\mu$. Our main tools are a general concentration inequality for determinantal point processes with a kernel that is a self-adjoint projection, and a strengthening of the Nevai condition from the theory of orthogonal polynomials.