The Annals of Probability

Circular law

Z. D. Bai

Full-text: Open access


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

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

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.

Export citation


  • BAI, Z. D. 1993a. Convergence rate of expected spectral distributions of large random matrices. I. Wigner matrices. Ann. Probab. 21 625 648. Z.
  • BAI, Z. D. 1993b. Convergence rate of expected spectral distributions of large random matrices. II. Sample covariance matrices. Ann. Probab. 21 649 672. Z.
  • BAI, Z. D. and YIN, Y. Q. 1986. Limiting behavior of the norm of products of random matrices and two problems of Geman-Hwang. Probab. Theory Related Fields 73 555 569. Z.
  • BAI, Z. D. and YIN, Y. Q. 1988a. A convergence to the semicircle law. Ann. Probab. 16 863 875. Z.
  • BAI, Z. D. and YIN, Y. Q. 1988b. Necessary and sufficient conditions for the almost sure convergence of the largest eigenvalue of Wigner matrices. Ann. Probab. 16 1729 1741. Z.
  • BAI, Z. D. and YIN, Y. Q. 1993. Limit of the smallest eigenvalue of large dimensional covariance matrix. Ann. Probab. 21 1275 1294. Z.
  • EDELMAN, A. 1995. The circular law and the probability that a random matrix has k real eigenvalues. Unpublished manuscript. Z.
  • GEMAN, S. 1980. A limit theorem for the norm of random matrices. Ann. Probab. 8 252 261. Z.
  • GEMAN, S. 1986. The spectral radius of large random matrices. Ann. Probab. 14 1318 1328. Z.
  • GINIBRE, J. 1965. Statistical ensembles of complex, quaterion and real matrices. J. Math. Phys. 440 449. Z.
  • GIRKO, V. L. 1984a. Circle law. Theory Probab. Appl. 694 706. Z.
  • GIRKO, V. L. 1984b. On the circle law. Theory Probab. Math. Statist. 15 23. Z.
  • HWANG, C. R. 1986. A brief survey on the spectral radius and the spectral distribution of large dimensional random matrices with iid entries. In Random Matrices and Their AppliZ. cations M. L. Mehta, ed. 145 152. Amer. Math. Soc., Providence, RI. Z.
  • JONSSON, D. 1982. Some limit theorems for the eigenvalues of a sample covariance matrix. J. Multivariate Anal. 12 1 38. Z.
  • LOEVE, M. 1977. Probability Theory, 4th ed. Springer, New York. Z.
  • MARCENKO, V. A. and PASTUR, L. A. 1967. Distribution for some sets of random matrices. Math. USSR-Sb. 1 457 483. Z.
  • PASTUR, L. A. 1972. On the spectrum of random matrices. Teoret. Mat. Fiz. 10 102 112. Z. English translation in Theoret. and Math. Phys. 10 67 74. Z.
  • PASTUR, L. A. 1973. Spectra of random self-adjoint operators. Uspekhi Mat. Nauk 28 4 63. Z. English translation in Russian Math. Surveys 28 1 67. Z.
  • SILVERSTEIN, J. W. and BAI, Z. D. 1995. On the empirical distribution of eigenvalues of a class of large dimensional random matrices. J. Multivariate Anal. 54 175 192.
  • SILVERSTEIN, W. J. and CHOI, S. I. 1995. Analysis of the limiting spectral distribution of large dimensional random matrices. J. Multivariate Anal. 54 295 309. Z.
  • WACHTER, K. W. 1978. The strong limits of random matrix spectra for sample matrices of independent elements. Ann. Probab. 6 1 18. Z.
  • WACHTER, K. W. 1980. The limiting empirical measure of multiple discriminant ratios. Ann. Statist. 8 937 957. Z.
  • YIN, Y. Q., BAI, Z. D. and KRISHNAIAH, P. R. 1988. On the limit of the largest eigenvalue of the large dimensional sample covariance matrix. Probab. Theory Related Fields 78 509 521.