Erdős László (IST Austria) Universality at Criticality: Cusp and circular edge

E. Wigner pioneering vision on the universality of the local statistics of eigenvalues of

large random matrices posed a major challenge for mathematicians. In the last decade

the celebrated Wigner--Dyson statistics in the bulk spectrum as well as the Tracy--Widom

statistics in the edge regime have been proven in great generality. In this talk I report on

the resolution of the last remaining universality regime that occurs at the cubic root cusps

in the density where the Pearcey statistics emerge. Understanding the cusp regime also

paved the way to prove edge universality for non-Hermitian matrices, a notoriously more

complicated ensemble than the Hermitian one.

The talk is based on joint works with G. Cipolloni, T. Kruger and D. Schroder.