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The Hermitian Two Matrix Model with an Even Quartic Potential
Maurice Duits, California Institute of Technology, Pasadena, CA, Arno B.J. Kuijlaars, Katholieke Universiteit Leuven, Belgium, and Man Yue Mo, University of Bristol, United Kingdom
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Memoirs of the American Mathematical Society
2012; 105 pp; softcover
Volume: 217
ISBN-10: 0-8218-6928-0
ISBN-13: 978-0-8218-6928-4
List Price: US$70 Individual Members: US$42
Institutional Members: US\$56
Order Code: MEMO/217/1022

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.

• Introduction and statement of results
• Preliminaries and the proof of Lemma 1.2
• Proof of Theorem 1.1
• A Riemann surface
• Pearcey integrals and the first transformation
• Second transformation $$X\mapsto U$$
• Opening of lenses
• Global parametrix
• Local parametrices and final transformation
• Bibliography
• Index