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The Hermitian two matrix model with an even quartic potential

About this Title

Maurice Duits, Department of Mathematics, California Institute of Technology, 1200 E. California Blvd, Pasadena California 91125, Arno B.J. Kuijlaars, Department of Mathematics, Katholieke Universiteit Leuven, Celestijnenlaan 200B, 3001 Leuven, Belgium and Man Yue Mo, Department of Mathematics, University of Bristol, Bristol BS8 1TW, United Kingdom

Publication: Memoirs of the American Mathematical Society
Publication Year: 2012; Volume 217, Number 1022
ISBNs: 978-0-8218-6928-4 (print); 978-0-8218-8756-1 (online)
Published electronically: September 20, 2011
Keywords:Two matrix model, eigenvalue distribution, correlation kernel, vector equilibrium problem, Riemann-Hilbert problem, steepest descent analysis.
MSC: Primary 30E25, 60B20; Secondary 15B52, 30F10, 31A05, 42C05, 82B26

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Table of Contents


  • Chapter 1. Introduction and statement of results
  • Chapter 2. Preliminaries and the proof of Lemma 1.2
  • Chapter 3. Proof of Theorem 1.1
  • Chapter 4. A Riemann surface
  • Chapter 5. Pearcey integrals and the first transformation
  • Chapter 6. Second transformation $X \mapsto U$
  • Chapter 7. Opening of lenses
  • Chapter 8. Global parametrix
  • Chapter 9. Local parametrices and final transformation


We 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 $M_1$. Our results generalize earlier results for the case $\alpha=0$, where the external field on the third measure was not present.

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