The Annals of Probability

The circular law for random matrices

Friedrich Götze and Alexander Tikhomirov

Full-text: Open access

Abstract

We consider the joint distribution of real and imaginary parts of eigenvalues of random matrices with independent entries with mean zero and unit variance. We prove the convergence of this distribution to the uniform distribution on the unit disc without assumptions on the existence of a density for the distribution of entries. We assume that the entries have a finite moment of order larger than two and consider the case of sparse matrices.

The results are based on previous work of Bai, Rudelson and the authors extending those results to a larger class of sparse matrices.

Article information

Source
Ann. Probab. Volume 38, Number 4 (2010), 1444-1491.

Dates
First available: 8 July 2010

Permanent link to this document
http://projecteuclid.org/euclid.aop/1278593956

Digital Object Identifier
doi:10.1214/09-AOP522

Mathematical Reviews number (MathSciNet)
MR2663633

Zentralblatt MATH identifier
1203.60010

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]

Keywords
Circular law random matrices

Citation

Götze, Friedrich; Tikhomirov, Alexander. The circular law for random matrices. The Annals of Probability 38 (2010), no. 4, 1444--1491. doi:10.1214/09-AOP522. http://projecteuclid.org/euclid.aop/1278593956.


Export citation

References

  • [1] Bai, Z. D. (1997). Circular law. Ann. Probab. 25 494–529.
  • [2] Bai, Z. D. and Silverstein, J. (2006). Spectral Analysis of Large Dimensional Random Matrices. Mathematics Monograph Series 2. Sciences Press, Beijing.
  • [3] Bickel, P. J., Götze, F. and van Zwet, W. R. (1986). The Edgeworth expansion for U-statistics of degree two. Ann. Statist. 14 1463–1484.
  • [4] Edelman, A. (1997). The probability that a random real Gaussian matrix has k real eigenvalues, related distributions, and the circular law. J. Multivariate Anal. 60 203–232.
  • [5] Friedland, S., Rider, B. and Zeitouni, O. (2004). Concentration of permanent estimators for certain large matrices. Ann. Appl. Probab. 14 1559–1576.
  • [6] Girko, V. L. (1989). Circular law. Theory Probab. Appl. 29 694–706.
  • [7] Ginibre, J. (1965). Statistical ensembles of complex, quaternion, and real matrices. J. Math. Phys. 6 440–449.
  • [8] Gohberg, I. C. and Krein, M. G. (1991). Introduction to the Theory of Linear Operator. Cambridge Univ. Press, New York.
  • [9] Götze, F. and Tikhomirov, A. (2003). Rate of convergence to the semi-circular law. Probab. Theory Related Fields 127 228–276.
  • [10] Götze, F. and Tikhomirov, A. On the circular law. Available at http://arxiv.org/abs/math/0702386.
  • [11] Horn, R. A. and Johnson, C. R. (1990). Matrix Analysis. Cambridge Univ. Press, Cambridge.
  • [12] Litvak, A. E., Pajor, A., Rudelson, M. and Tomczak-Jaegermann, N. (2005). Smallest singular value of random matrices and geometry of random polytopes. Adv. Math. 195 491–523.
  • [13] Pan, G. and Zhou, W. (2010). Circular law, extreme singular values and potential theory. J. Multivariate Anal. 101 645–656.
  • [14] Saff, E. B. and Totik, V. (1997). Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 316. Springer, Berlin.
  • [15] Mehta, M. L. (1991). Random Matrices, 2nd ed. Academic Press, Boston, MA.
  • [16] Pastur, L. A. (1973). Spectra of random selfadjoint operators. Uspehi Mat. Nauk 28 3–64.
  • [17] Rudelson, M. (2008). Invertibility of random matrices: Norm of the inverse. Ann. of Math. (2) 168 575–600.
  • [18] Rudelson, M. and Vershynin, R. (2008). The Littlewood–Offord problem and invertibility of random matrices. Adv. Math. 218 600–633.
  • [19] Petrov, V. V. (1975). Sums of Independent Random Variables. Springer, New York.
  • [20] Rider, B. and Virág, B. (2007). The noise in the circular law and the Gaussian free field. Int. Math. Res. Not. IMRN 2 33.
  • [21] Rider, B. (2003). A limit theorem at the edge of a non-Hermitian random matrix ensemble: Random matrix theory. J. Phys. A 36 3401–3409.
  • [22] Tao, T. and Vu, V. (2008). Random matrices: The circular law. Commun. Contemp. Math. 10 261–307.
  • [23] Tao, T. and Vu, V. H. (2009). Inverse Littlewood–Offord theorems and the condition number of random discrete matrices. Ann. of Math. (2) 169 595–632.
  • [24] Timme, M., Wolf, F. and Geisel, T. (2004). Topological speed limits to network synchronization. Phys. Rev. Lett. 92 074101-1–4.