Mean eigenvector self-overlap in deformed complex Ginibre ensemble

Consider a random matrix of size N as an additive deformation of the complex Ginibre ensemble under a deterministic matrix X₀ with a finite rank, independent of N. We prove that microscopic statistics for the mean diagonal overlap, near the edge point, are characterized by the iterative erfc integrals, which only depend on the geometric multiplicity of certain eigenvalue of X₀. Further we get a phenomenon of the phase transition for the mean diagonal overlap, with respect to the modulus of the eigenvalues of X₀.