Large n limit of Gaussian random matrices with external source, part I

TL;DR

This paper studies the asymptotic behavior of eigenvalues in a Gaussian random matrix ensemble with an external source, revealing universal local correlation patterns similar to classical models when the external source parameter exceeds one.

Contribution

It establishes the universal local eigenvalue correlation behavior for large matrices with an external source, using Riemann-Hilbert analysis of multiple Hermite polynomials.

Findings

Eigenvalue correlations follow sine kernel in the bulk.

Airy kernel describes edge eigenvalue behavior.

Results hold for external source parameter a > 1.

Abstract

We consider the random matrix ensemble with an external source \[ \frac{1}{Z_n} e^{-n \Tr({1/2}M^2 -AM)} dM \] defined on $n\times n$ Hermitian matrices, where $A$ is a diagonal matrix with only two eigenvalues $\pm a$ of equal multiplicity. For the case $a > 1$, we establish the universal behavior of local eigenvalue correlations in the limit $n \to \infty$, which is known from unitarily invariant random matrix models. Thus, local eigenvalue correlations are expressed in terms of the sine kernel in the bulk and in terms of the Airy kernel at the edge of the spectrum. We use a characterization of the associated multiple Hermite polynomials by a $3 \times 3$-matrix Riemann-Hilbert problem, and the Deift/Zhou steepest descent method to analyze the Riemann-Hilbert problem in the large $n$ limit.