Universality of the double scaling limit in
multi-critical random matrix ensembles

Tom Claeys

Abstract: We study unitary random matrix ensembles with a hard singularity at the origin,
coinciding with a soft singularity due to quadratic vanishing of the limiting mean eigenvalue density.
We establish universality of the eigenvalue correlation kernel near the origin in a double scaling limit.
This involves functions linked to a special solution of the general Painlevé II equation. In addition,
we show that this Painlevé II function, which is a generalization of the Hastings-McLeod solution,
has no real poles.

(Joint work with Arno Kuijlaars and Maarten Vanlessen)