# The Hermitian Two Matrix Model with an Even Quartic Potential

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*Maurice Duits; Arno B.J. Kuijlaars; Man Yue Mo*

The authors 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 \(4\times4\) matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of \(M_1\). The authors' results generalize earlier results for the case \(\alpha=0\), where the external field on the third measure was not present.

#### Table of Contents

# Table of Contents

## The Hermitian Two Matrix Model with an Even Quartic Potential

- Chapter 1. Introduction and Statement of Results 18 free
- Chapter 2. Preliminaries and the Proof of Lemma 1.2 1522
- Chapter 3. Proof of Theorem 1.1 2128
- Chapter 4. A Riemann Surface 3138
- Chapter 5. Pearcey Integrals and the First Transformation 4148
- Chapter 6. Second Transformation X U 5360
- Chapter 7. Opening of Lenses 6168
- Chapter 8. Global Parametrix 7582
- 8.1. Statement of RH problem 7582
- 8.2. Riemann surface as an M-curve 7683
- 8.3. Canonical homology basis 7784
- 8.4. Meromorphic differentials 7986
- 8.5. Definition and properties of functions uj 8188
- 8.6. Definition and properties of functions vj 8491
- 8.7. The first row of M 8592
- 8.8. The other rows of M 8693

- Chapter 9. Local Parametrices and Final Transformation 8996
- Bibliography 99106
- Index 103110 free