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 $$A+X$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>A</mml:mi> <mml:mo>+</mml:mo> <mml:mi>X</mml:mi> </mml:mrow> </mml:math> 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 (for normal deformations), after bulk and sharp edge universalities have been established in recent years.