# The Hermitian Two Matrix Model with an Even Quartic Potential

American Mathematical Soc., 2012 - Boundary value problems - 105 pages
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.

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### Contents

 Chapter 1 Introduction and Statement of Results 1 Chapter 2 Preliminaries and the Proof of Lemma 12 15 Chapter 3 Proof of Theorem 11 21 Chapter 4 A Riemann Surface 31 Chapter 5 Pearcey Integrals and the First Transformation 41 Chapter 6 Second Transformation X U 53
 Chapter 7 Opening of Lenses 61 Chapter 8 Global Parametrix 75 Chapter 9 Local Parametrices and Final Transformation 89 Bibliography 99 Index 103 Copyright