The Hermitian two matrix model with an even quartic potential

TL;DR

This paper analyzes the eigenvalue distribution of a two matrix model with an even quartic potential, formulating a vector equilibrium problem and employing Riemann-Hilbert analysis to generalize previous results.

Contribution

It introduces a new vector equilibrium formulation for the eigenvalue distribution in the two matrix model with an even quartic potential, extending prior work by including an external field.

Findings

Formulation of a vector equilibrium problem for eigenvalues

Use of Riemann-Hilbert steepest descent analysis

Generalization of earlier results for alpha=0

Abstract

We consider the two matrix model with an even quartic potential W(y)=y^4/4+alpha y^2/2 and an even polynomial potential V(x). The main result of the paper is the formulation of a vector equilibrium problem for the limiting mean density for the eigenvalues of one of the matrices M_1. The vector equilibrium problem is defined for three measures, with external fields on the first and third measures and an upper constraint on the second measure. The proof is based on a steepest descent analysis of a 4 x 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M_1. Our results generalize earlier results for the case alpha=0, where the external field on the third measure was not present.