Large n-limit for Random matrices with External Source with 3 eigenvalues

TL;DR

This paper investigates the large n-limit behavior of Hermite random matrices with an external source having three eigenvalues, establishing universal local eigenvalue correlations in the limit.

Contribution

It demonstrates the universal local eigenvalue correlation behavior for Hermite matrices with three external eigenvalues when a^2>3, using Riemann-Hilbert analysis.

Findings

Eigenvalue correlations follow sine and Airy kernels in the bulk and at the spectrum edge.

Universal behavior matches known results from unitarily invariant models.

Analysis performed via 4x4 Riemann-Hilbert problem and nonlinear steepest descent.

Abstract

In this paper, we analyze the large n-limit for random matrix with external source with three distinct eigenvalues. And we confine ourselves in the Hermite case and the three distinct eigenvalues are $-a,0,a$. For the case $a^2>3$, we establish the universal behavior of local eigenvalue correlations in the limit $n\rightarrow \infty$, 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. The result can be obtained by analyzing $4\times 4$ Riemann-Hilbert problem via nonlinear steepest decent method.