Asymptotic corrections to the eigenvalue density of the GUE and LUE

TL;DR

This paper derives precise correction terms for the eigenvalue density of GUE and LUE random matrix ensembles in large N limits, improving understanding of spectral behavior in bulk and near edges.

Contribution

It introduces a method to compute asymptotic correction terms for eigenvalue densities using contour integrals and saddle point analysis, applicable to GUE and LUE.

Findings

Correction terms are oscillatory in N in the bulk.

Edge corrections are expressed via Airy functions.

Numerical results confirm high accuracy of expansions.

Abstract

We obtain correction terms to the large N asymptotic expansions of the eigenvalue density for the Gaussian unitary and Laguerre unitary ensembles of random N by N matrices, both in the bulk of the spectrum and near the spectral edge. This is achieved by using the well known orthogonal polynomial expression for the kernel to construct a double contour integral representation for the density, to which we apply the saddle point method. The main correction to the bulk density is oscillatory in N and depends on the distribution function of the limiting density, while the corrections to the Airy kernel at the soft edge are again expressed in terms of the Airy function and its first derivative. We demonstrate numerically that these expansions are very accurate. A matching is exhibited between the asymptotic expansion of the bulk density, expanded about the edge, and the asymptotic expansion of the edge density, expanded into the bulk.