Non-Hermitian spectral universality at critical points

TL;DR

This paper proves that the local eigenvalue statistics near critical points of large non-Hermitian random matrices are universal, completing the classification of universality classes for such matrices.

Contribution

It establishes the universality of eigenvalue statistics at critical points for a broad class of non-Hermitian matrices, filling a key gap in the theory.

Findings

Universality at critical spectral points proven for non-Hermitian matrices

Completes the classification of spectral universality classes

Extends previous bulk and edge universality results

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.