Circular law for non-central random matrices

Abstract

Let $(X_{jk})_{j,k\geq 1}$ 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 $(\frac{1}{\sqrt{n}}X_{jk})_{1\leq j,k\leq n}$. 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 $n\to\infty$ 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 $M$ to $(X_{jk})_{1\leq j,k\leq n}$ provided that $\mathrm{Tr}(MM^*)=O(n^2)$ 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.