Eigenvalue Separation in Some Random Matrix Models

TL;DR

This paper investigates eigenvalue separation phenomena in various random matrix models, analyzing conditions under which eigenvalues detach from the bulk spectrum using asymptotic and exact methods.

Contribution

It provides a comparative analysis of eigenvalue separation in Gaussian, Wishart, and chiral ensembles, employing asymptotic analysis and exact density formulas.

Findings

Eigenvalue separation occurs when the shift c exceeds 1 in the large N limit.

Exact density formulas reveal detailed eigenvalue behavior and separation effects.

Comparison across models highlights universal and model-specific eigenvalue phenomena.

Abstract

The eigenvalue density for members of the Gaussian orthogonal and unitary ensembles follows the Wigner semi-circle law. If the Gaussian entries are all shifted by a constant amount c/Sqrt(2N), where N is the size of the matrix, in the large N limit a single eigenvalue will separate from the support of the Wigner semi-circle provided c > 1. In this study, using an asymptotic analysis of the secular equation for the eigenvalue condition, we compare this effect to analogous effects occurring in general variance Wishart matrices and matrices from the shifted mean chiral ensemble. We undertake an analogous comparative study of eigenvalue separation properties when the size of the matrices are fixed and c goes to infinity, and higher rank analogues of this setting. This is done using exact expressions for eigenvalue probability densities in terms of generalized hypergeometric functions, and using the interpretation of the latter as a Green function in the Dyson Brownian motion model. For the shifted mean Gaussian unitary ensemble and its analogues an alternative approach is to use exact expressions for the correlation functions in terms of classical orthogonal polynomials and associated multiple generalizations. By using these exact expressions to compute and plot the eigenvalue density, illustrations of the various eigenvalue separation effects are obtained.