Circular law for non-central random matrices

Djalil Chafai (1)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées

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 $\la_{n,1},\ldots,\la_{n,n}$ 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 $\frac{1}{n}(\de_{\la_{n,1}}+\cdots+\de_{\la_{n,n}})$ 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.

Mots clés: Random matrices; Circular law

Domaines: Probabilités [math.PR]

Djalil Chafai. Circular law for non-central random matrices. Journal of Theoretical Probability, 2010, 23 (4), pp.945-950. ⟨10.1007/s10959-010-0285-8⟩. ⟨hal-00169167v3⟩