Circular law for non-central random matrices

TL;DR

This paper extends the circular law for non-central random matrices by providing an elementary proof that allows adding a deterministic matrix with controlled trace and rank, broadening the law's applicability.

Contribution

It introduces a simple argument to incorporate deterministic matrices into the circular law framework under specific trace and rank conditions.

Findings

The empirical spectral distribution converges to the uniform law over the unit disc.

The method applies to matrices with added deterministic components with bounded trace and rank.

The approach parallels techniques used for non-central Wigner and Marchenko-Pastur theorems.

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.