Critical edge statistics for deformed GinUEs

For the complex Ginibre ensemble subjected to an additive perturbation by a deterministic normal matrix $X_0$, we establish that under specific spectral conditions on $X_0$, only two distinct types of local spectral statistics emerge at the spectral edge: GinUE statistics and critical statistics, which respectively correspond to regular and quadratically vanishing spectral points. The critical statistics, as a non-Hermitian analogue of Pearcey statistics in random matrix theory, describes a novel point process on the complex plane. This identifies the third (and likely final) universal statistics in non-Hermitian random matrix theory, after the established GinUE bulk and edge universality classes, and represents the primary achievement of this paper.