The Hermitian two matrix model with an even quartic potential

Abstract

We consider the two matrix model with an even quartic potential W(y) = y^4/4 + αy^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 x 4 matrix valued Riemann-Hilbert problem that characterizes the correlation kernel for the eigenvalues of M_1. Our results generalize earlier results for the case α = 0, where the external field on the third measure was not present.

Additional Information

© 2011 American Mathematical Society. Received by editor(s): October 20, 2010; Posted: September 20, 2011. M. Duits and A.B.J. Kuijlaars are grateful for the support and hospitality of MSRI in Berkeley in the fall of 2010. A.B.J. Kuijlaars is supported by K.U. Leuven research grant OT/08/33, FWO-Flanders project G.0427.09 and G.0641.11, by the Belgian Interuniversity Attraction Pole P06/02, and by grant MTM2008-06689-C02-01 of the Spanish Ministry of Science and Innovation. M. Y. Mo acknowledges financial support by the EPSRC grant EP/G019843/1.

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Published - Duits2012p19373Mem_Am_Math_Soc.pdf

Submitted - 1010.4282v1.pdf

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