Journal of Theoretical Probability

, Volume 23, Issue 4, pp 945–950

Circular Law for Noncentral Random Matrices

Authors

    • Laboratoire d’Analyse et de Mathématiques AppliquéesUMR 8050 CNRS, Université Paris-Est Marne-la-Vallée
Article

DOI: 10.1007/s10959-010-0285-8

Cite this article as:
Chafaï, D. J Theor Probab (2010) 23: 945. doi:10.1007/s10959-010-0285-8

Abstract

Let (Xjk)j,k1 be an infinite array of i.i.d. complex random variables with mean 0 and variance 1. Let λn,1,…,λn,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 (Xjk)1j,kn provided that Tr(MM*)=O(n2) and rank(M)=O(nα) with α<1. Conveniently, the argument is similar to the one used for the noncentral version of the Wigner and Marchenko–Pastur theorems.

Keywords

Random matricesCircular law

Mathematics Subject Classification (2000)

15B52

Copyright information

© Springer Science+Business Media, LLC 2010