  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 $N$. 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 $X_0$. Further we get a phenomenon of the phase transition for
the mean diagonal overlap, with respect to the modulus of the eigenvalues of
$X_0$.
