A revisit of the circular law

TL;DR

This paper revisits the proof of the circular law for random matrices, extending its validity under Lindeberg's condition and introducing new theoretical tools to simplify the proof process.

Contribution

It extends the circular law to matrices satisfying Lindeberg's condition and simplifies the proof using new strong law of large numbers and eigenvalue bounds.

Findings

Circular law holds under Lindeberg's condition.

Established a strong law of large numbers for eigenvalues.

Provided a simplified proof approach for the circular law.

Abstract

Consider a complex random $n\times n$ matrix ${\bf X}_n=(x_{ij})_{n\times n}$, whose entries $x_{ij}$ are independent random variables with zero means and unit variances. It is well-known that Tao and Vu (Ann Probab 38: 2023-2065, 2010) resolved the circular law conjecture, establishing that if the $x_{ij}$'s are independent and identically distributed random variables with zero mean and unit variance, the empirical spectral distribution of $\frac{1}{\sqrt{n}}{\bf X}_n$ converges almost surely to the uniform distribution over the unit disk in the complex plane as $n \to \infty$. This paper demonstrates that the circular law still holds under the more general Lindeberg's condition: $$ \frac1{n^2}\sum_{i,j=1}^n\mathbb{E}|x_{ij}^2|I(|x_{ij}|>\eta\sqrt{n})\to 0,\mbox{as $n \to \infty$}. $$ This paper is a revisit of the proof procedure of the circular law by Bai in (Ann Probab 25: 494-529, 1997). The key breakthroughs in the paper are establishing a general strong law of large numbers under Lindeberg's condition and the uniform upper bound for the integral with respect to the smallest eigenvalues of random matrices. These advancements significantly streamline and clarify the proof of the circular law, offering a more direct and simplified approach than other existing methodologies.