The Annals of Probability

Circular law

Z. D. Bai

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It was conjectured in the early 1950’s that the empirical spectral distribution of an $n \times n$ matrix, of iid entries, normalized by a factor of $\frac{1}{\sqrt{n}}$, converges to the uniform distribution over the unit disc on the complex plane, which is called the circular law. Only a special case of the conjecture, where the entries of the matrix are standard complex Gaussian, is known. In this paper, this conjecture is proved under the existence of the sixth moment and some smoothness conditions. Some extensions and discussions are also presented.

Article information

Ann. Probab., Volume 25, Number 1 (1997), 494-529.

First available in Project Euclid: 18 June 2002

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Zentralblatt MATH identifier

Primary: 60F15: Strong theorems
Secondary: 62H99: None of the above, but in this section

Circular law complex random matrix noncentral Hermitian matrix largest and smallest eigenvalue of random matrix spectral radius spectral analysis of large-dimensional random matrices


Bai, Z. D. Circular law. Ann. Probab. 25 (1997), no. 1, 494--529. doi:10.1214/aop/1024404298.

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