Circular law for non-central random matrices

Abstract

Let $(X_{jk}{)}_{j,k\ge 1}$ 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 $(\frac{1}{\sqrt{n}}{X}_{jk}{)}_{1\le j,k\le n}$. 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 $(X_{jk}{)}_{1\le j,k\le n}$ provided that $\mathrm{Tr}(M{M}^{\ast })=O(n^2)$ 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.

accepted in Journal of Theoretical Probability

References in corpus (7)

• The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations
• The circular law for random matrices
• The Dirichlet Markov Ensemble
• Random matrices: Universality of ESDs and the circular law
• Random Matrices: The circular Law
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Cited by in corpus (11)

• Around the circular law
• Circular Law Theorem for Random Markov Matrices
• Spectrum of non-Hermitian heavy tailed random matrices
• Circular law for random matrices with exchangeable entries
• The Dirichlet Markov Ensemble
• Spectrum of Markov generators on sparse random graphs
• Universal sum and product rules for random matrices
• Random matrices: Universality of ESDs and the circular law
• Spectrum of large random reversible Markov chains: two examples
• Outliers in the spectrum of iid matrices with bounded rank perturbations
• Universality of the ESD for a fixed matrix plus small random noise: a stability approach