Universality of the matrix Airy and Bessel functions at spectral edges of unitary ensembles

TL;DR

This paper establishes the universal behavior of matrix Airy and Bessel functions at spectral edges of unitary ensembles, linking them to matrix models and extending universality results.

Contribution

It proves universality at the soft edge and expresses the limits using Kontsevich matrix models, connecting spectral edge behavior to universal matrix functions.

Findings

Universality at the soft edge of eigenvalue density.

Universal limits expressed via matrix Airy and Bessel functions.

Connection to Kontsevich matrix models for spectral edges.

Abstract

This paper deals with products and ratios of average characteristic polynomials for unitary ensembles. We prove universality at the soft edge of the limiting eigenvalues' density, and write the universal limit in function of the Kontsevich matrix model ("matrix Airy function", as originally named by Kontsevich). For the case of the hard edge, universality is already known. We show that also in this case the universal limit can be expressed as a matrix integral ("matrix Bessel function") known in the literature as generalized Kontsevich matrix model.