Asymptotic form of the density profile for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry

TL;DR

This paper extends the asymptotic analysis of eigenvalue densities to Gaussian and Laguerre ensembles with orthogonal and symplectic symmetry, revealing matching expansions and microscopic interpretations.

Contribution

It provides the first detailed asymptotic expansions for these ensembles, including correction terms and microscopic insights, building on previous work for unitary ensembles.

Findings

Matching between bulk and edge density expansions

Asymptotic correction terms derived for orthogonal and symplectic cases

Microscopic interpretation of density expansions involving delta functions

Abstract

In a recent study we have obtained 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 and at the soft edge of the spectrum. In the present study these results are used to similarly analyze the eigenvalue density for Gaussian and Laguerre random matrix ensembles with orthogonal and symplectic symmetry. As in the case of unitary symmetry, 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. In addition, aspects of the asymptotic expansion of the smoothed density, which involves delta functions at the endpoints of the support, are interpreted microscopically.