Large n limit of Gaussian random matrices with external source, Part III: Double scaling limit

TL;DR

This paper investigates the double scaling limit of Gaussian random matrices with an external source at a critical value, revealing new universal eigenvalue correlation behavior described by Pearcey integrals.

Contribution

It extends the analysis of eigenvalue correlations at the critical point by constructing a Pearcey integral-based local parametrix using Riemann-Hilbert techniques.

Findings

Identifies Pearcey integral behavior at the critical point a=1

Develops a Riemann-Hilbert approach for the double scaling limit

Matches local Pearcey parametrix with global solution successfully

Abstract

We consider the double scaling limit in 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 two eigenvalues $\pm a$ of equal multiplicities. The value $a=1$ is critical since the eigenvalues of $M$ accumulate as $n \to \infty$ on two intervals for $a > 1$ and on one interval for $0 < a < 1$. These two cases were treated in Parts I and II, where we showed that the local eigenvalue correlations have the universal limiting behavior known from unitary random matrix ensembles. For the critical case $a=1$ new limiting behavior occurs which is described in terms of Pearcey integrals, as shown by Br\'ezin and Hikami, and Tracy and Widom. We establish this result by applying the Deift/Zhou steepest descent method to a $3 \times 3$-matrix valued Riemann-Hilbert problem which involves the construction of a local parametrix out of Pearcey integrals. We resolve the main technical issue of matching the local Pearcey parametrix with a global outside parametrix by modifying an underlying Riemann surface.