The Hermitian Two Matrix Model with an Even Quartic Potential

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