Maurice Duits: Painlevé kernels in Hermitian matrix models

The universality principle in random matrix theory states that the local eigenvalues statistics of large random matrices obey laws that have a universal character and do not depend on the precise definition of the underlying probability measure. Generic examples are the sine universality in the bulk and Airy universality near a soft edge. In special situations the spectral curve has singular points and if these points are on the physical sheet we obtain more complicated universality classes. In some of these cases remarkable connections to the Painlevé equations have been found. In this talk I will start by reviewing the singular cases that appear in the Hermitian one matrix model. Afterwards, I will discuss the recent progress on the Hermitian two matrix model. In particular, I will discuss a new limiting kernel in the quartic/quadratic case that is constructed out of a 4x4 Riemann-Hilbert problem related to Painlevé II equation.