Non-Hermitian spectral universality at critical points

Abstract: For general large non-Hermitian random matrices $X$ and deterministic normal deformations $A$, we prove that the local eigenvalue statistics of $A+X$ close to the critical edge points of its spectrum are universal. This concludes the proof of the third and last remaining typical universality class for non-Hermitian random matrices, after bulk and sharp edge universalities have been established in recent years.