Circular law for non-central random matrices

Let (Xi,j)16i,j<∞ be an infinite array of i.i.d. complex random variables, with mean m = 0, variance σ2 = 1, and say with finite fourth moment. The famous circular law theorem states that the empirical spectral distribution 1 n(δλ1(X)+· · ·+δλn(X)) of X = (n−1/2Xi,j)16i,j6n converges almost surely, as n → ∞, to the uniform law over the unit disc {z ∈ C; |z| 6 1}. For now, most efforts where focused on the improvement of moments hypotheses for the centered case m = 0. Regarding the non-central case m 6= 0, Silverstein has already observed that almost surely, the eigenvalue of X of largest module goes to infinity as n → ∞, while the rest of the spectrum remains bounded. We show that the circular law theorem remains valid when m 6= 0, by using logarithmic potentials and bounds on extremal singular values. AMS 2000 Mathematical Subject Classification: 60F15; 15A52; 62H99.

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