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January 1997 Circular law
Z. D. Bai
Ann. Probab. 25(1): 494-529 (January 1997). DOI: 10.1214/aop/1024404298


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.


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Z. D. Bai. "Circular law." Ann. Probab. 25 (1) 494 - 529, January 1997.


Published: January 1997
First available in Project Euclid: 18 June 2002

zbMATH: 0871.62018
MathSciNet: MR1428519
Digital Object Identifier: 10.1214/aop/1024404298

Primary: 60F15
Secondary: 62H99

Keywords: circular law , complex random matrix , largest and smallest eigenvalue of random matrix , noncentral Hermitian matrix , spectral analysis of large-dimensional random matrices , spectral radius

Rights: Copyright © 1997 Institute of Mathematical Statistics


Vol.25 • No. 1 • January 1997
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