, Volume 23, Issue 4, pp 945-950
Date: 22 Apr 2010

Circular Law for Noncentral Random Matrices

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Abstract

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<1. Conveniently, the argument is similar to the one used for the noncentral version of the Wigner and Marchenko–Pastur theorems.