## The Annals of Probability

### Circular law

Z. D. Bai

#### Abstract

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

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

Dates
First available in Project Euclid: 18 June 2002

http://projecteuclid.org/euclid.aop/1024404298

Digital Object Identifier
doi:10.1214/aop/1024404298

Mathematical Reviews number (MathSciNet)
MR1428519

Zentralblatt MATH identifier
0871.62018

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

#### Citation

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

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• KAOHSIUNG, TAIWAN 80424 E-MAIL: baiz@math.nsysu.edu.tw