Painlevé kernels in Hermitian matrix models

Abstract

After reviewing the Hermitian one matrix model, we will give a brief introduction to the Hermitian two matrix model and present a summary of some recent results on the asymptotic behavior of the two matrix model with a quartic potential. In particular, we will discuss a limiting kernel in the quartic/quadratic case that is constructed out of a $4\times 4$ Riemann-Hilbert problem related to Painlevé II equation. Also an open problem will be presented.

26 pages, 5 figures; This is a contribution to the Special Issue on Painlevé equations in the journal Constructive Approximation

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