The Hermitian two matrix model with an even quartic potential

2010/10/20 by Maurice Duits, Arno B. J. Kuijlaars, Duits, Maurice +3
Mathematics · Physics and Astronomy · #30E25 #60B20 #Complex Variables (math.CV) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Probability (math.PR) #math-ph #math.CV #math.MP #math.PR #msc:30E25 #msc:60B20

paper · pdf · doi:10.48550/arxiv.1010.4282

123 pages, 15 figures

arxiv created 2010/10/20 · arxiv updated 2010/10/21

Abstract

We consider the two matrix model with an even quartic potential W(y)=y⁴/4+alpha y²/2 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 M₁. 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 x 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M₁. 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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