A revisit of 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.