Circular law for non-central random matrices


Let (Xjk)j,k1(X_{jk})_{j,k\geq 1} be an infinite array of i.i.d. complex random variables, with mean 0 and variance 1. Let \la_{n,1},...,\la_{n,n} be the eigenvalues of (1nXjk)1j,kn(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}. The strong circular law theorem states that with probability one, the empirical spectral distribution \frac{1}{n}(\de_{\la_{n,1}}+...+\de_{\la_{n,n}}) converges weakly as nn\to\infty to the uniform law over the unit disc \{z\in\dC;|z|\leq1\}. In this short note, we provide an elementary argument that allows to add a deterministic matrix MM to (Xjk)1j,kn(X_{jk})_{1\leq j,k\leq n} provided that Tr(MM)=O(n2)\mathrm{Tr}(MM^*)=O(n^2) and \mathrm{rank}(M)=O(n^\al) with \al<1. Conveniently, the argument is similar to the one used for the non-central version of Wigner's and Marchenko-Pastur theorems.Comment: accepted in Journal of Theoretical Probabilit

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