Circular Law for Noncentral Random Matrices

Let (X jk )j,k≥1 be an infinite array of i.i.d. complex random variables with mean 0 and variance 1. Let λ n,1,…,λ n,n be the eigenvalues of $(\frac{1}{\sqrt{n}}X_{jk})_{1\leqslant j,k\leqslant n}$ . The strong circular law theorem states that, with probability one, the empirical spectral distribution $\frac{1}{n}(\delta _{\lambda _{n,1}}+\cdots+\delta _{\lambda _{n,n}})$ converges weakly as n→∞ to the uniform law over the unit disc {z∈ℂ,|z|≤1}. In this short paper, we provide an elementary argument that allows us to add a deterministic matrix M to (X jk )1≤ j,k ≤ n provided that Tr(MM *)=O(n 2) and rank(M)=O(n α ) with α