The largest eigenvalue of real symmetric, Hermitian and Hermitian self-dual random matrix models with rank one external source, part I

TL;DR

This paper studies the asymptotic behavior of the largest eigenvalue in various real symmetric and Hermitian random matrix models with a rank-one external source, providing a unified analysis under certain regularity conditions.

Contribution

It introduces a contour integral approach to analyze the largest eigenvalue distribution across different matrix ensembles with external sources, covering non-critical cases.

Findings

Limiting location of the largest eigenvalue for non-critical external source eigenvalues.

Limiting distribution of the largest eigenvalue when the external source eigenvalue exceeds the critical value.

Unified analysis method applicable to all $eta$ > 0 matrix models.

Abstract

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric ($\beta$ = 1), Hermitian ($\beta$ = 2), and Hermitian self-dual ($\beta$ = 4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the one-band condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all $\beta$ > 0.