Consider a random matrix of size N as an additive deformation of the complex Ginibre ensemble under a deterministic matrix \(X_0\) with a finite rank, independent of N. When some eigenvalues of \(X_0\) separate from the unit disk, outlier eigenvalues may appear asymptotically in the same locations, and their fluctuations exhibit surprising phenomena that highly depend on the Jordan canonical form of \(X_0\) . These findings are largely due to Benaych-Georges and Rochet (Probab. Theory Relat. Fields, 165:313–363, 2016), Bordenave and Capitaine (Comm. Pure Appl. Math. 69:2131–2194, 2016), and Tao (Probab. Theory Relat. Fields 155:231–263, 2013). Yet when there is an eigenvalue of \(X_0\) on the edge of the unit disk, we prove that local eigenvalue statistics at the same spectral edge form a new class of determinantal point processes, for which correlation kernels only depend on geometric multiplicity of eigenvalue and are characterized in terms of the iterated erfc functions. This thus completes a non-Hermitian analogue of the BBP phase transition in Random Matrix Theory.
