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

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 |

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