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Circular law, extreme singular values and potential theory

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  • Pan, Guangming
  • Zhou, Wang

Abstract

Consider the empirical spectral distribution of complex random nxn matrix whose entries are independent and identically distributed random variables with mean zero and variance 1/n. In this paper, via applying potential theory in the complex plane and analyzing extreme singular values, we prove that this distribution converges, with probability one, to the uniform distribution over the unit disk in the complex plane, i.e. the well known circular law, under the finite fourth moment assumption on matrix elements.

Suggested Citation

  • Pan, Guangming & Zhou, Wang, 2010. "Circular law, extreme singular values and potential theory," Journal of Multivariate Analysis, Elsevier, vol. 101(3), pages 645-656, March.
    • Handle: RePEc:eee:jmvana:v:101:y:2010:i:3:p:645-656
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    References listed on IDEAS

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    1. Bai, Z. D. & Silverstein, Jack W. & Yin, Y. Q., 1988. "A note on the largest eigenvalue of a large dimensional sample covariance matrix," Journal of Multivariate Analysis, Elsevier, vol. 26(2), pages 166-168, August.
    2. Edelman, Alan, 1997. "The Probability that a Random Real Gaussian Matrix haskReal Eigenvalues, Related Distributions, and the Circular Law," Journal of Multivariate Analysis, Elsevier, vol. 60(2), pages 203-232, February.
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    Cited by:

    1. Sean O’Rourke & David Renfrew, 2016. "Central Limit Theorem for Linear Eigenvalue Statistics of Elliptic Random Matrices," Journal of Theoretical Probability, Springer, vol. 29(3), pages 1121-1191, September.
    2. Natalie Coston & Sean O’Rourke, 2020. "Gaussian Fluctuations for Linear Eigenvalue Statistics of Products of Independent iid Random Matrices," Journal of Theoretical Probability, Springer, vol. 33(3), pages 1541-1612, September.
    3. Djalil Chafaï, 2010. "Circular Law for Noncentral Random Matrices," Journal of Theoretical Probability, Springer, vol. 23(4), pages 945-950, December.

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