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We study the deformed complex Ginibre ensemble H=A₀+H₀, where H₀ is the complex matrix with iid Gaussian entries, and A₀ is some general n n matrix (it can be random and in this case it is independent of H₀). Assuming rather general assumptions on A₀, we prove that the asymptotic local behavior of the second correlation function of the eigenvalues of such matrices in the bulk coincides with that for the pure complex Ginibre ensemble.
Afanasiev et al. (Wed,) studied this question.