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The Hermitian two matrix model with an even quartic potential

Authors: Maurice Duits, Arno B.J. Kuijlaars and Man Yue Mo
Journal: Memoirs of the AMS 217 (2012)
MSC (2010): Primary 30E25, 60B20; Secondary 15B52, 30F10, 31A05, 42C05, 82B26
Posted: September 20, 2011
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Abstract: 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 . 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 . 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 . 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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