Skip to main content
SpringerLink
Log in

Asymptotic distribution of singular values for matrices in a spherical ensemble

  • Published:
Siberian Advances in Mathematics Aims and scope Submit manuscript

Abstract

We consider the asymptotic behavior of the singular values of a so-called spherical ensemble of random matrices of large dimension. These are matrices of the form XY −1, where X and Y are independent matrices of dimension n × n whose symmetric entries have correlation coefficient ρ. We show that the limit distribution of the singular values is independent of the correlation coefficient and has the density

$$p(x) = \frac{1} {{\pi \sqrt x (1 + x)}}\mathbb{I}\{ x > 0\} ,$$

where \(\mathbb{I}\{ A\}\) stands for the indicator of an event A.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. A. Alastuey A. and B. Jancovici, “On the classical two-dimensional one-component Coulomb plasma,” J. Physique 42, 1 (1981).

    Article  MathSciNet  Google Scholar 

  2. Z. D. Bai, “Circular law,” Ann. Probab. 25, 494 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  3. Z. D. Bai and J. Silverstein, Spectral Analysis of Large Dimensional Random Matrices. 2nd ed. (Springer, New York, 2009).

    Google Scholar 

  4. Ch. Bordenave, “On the spectrum of sum and product of non-Hermitian random matrices,” Electron. Comm. Probab. 16, 104 (2011).

    Article  MATH  MathSciNet  Google Scholar 

  5. Ch. Bordenave and D. Chafaï, Around the Circular Law, Preprint arXiv: 1109.3343 (2012).

    Google Scholar 

  6. A. Edelman, E. Kostlan, and M. Shub, “How many eigenvalues of a random matrix are real?” J. Amer. Math. Soc. 7, 247 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  7. J. Fischmann and P. Forrester, “One-component plasma on a spherical annulus and a random matrix ensemble,” J. Stat. Mech.: Theory and Experiment. P10003 DOI: 10.1088/1742-5468/2011/10/P10003 (2011).

    Google Scholar 

  8. P. Forrester and A. Mays, “Pfaffian point process for the Gaussian real generalized eigenvalue problem,” Probab. Theory Related Fields, 154, 1 (2012).

    Article  MATH  MathSciNet  Google Scholar 

  9. Ya. V. Fyodorov, B. A. Khoruzhenko, and H. Y. Sommers, “Almost-Hermitian random matrices: Eigenvalue density in the complex plane,” Phys. Lett., A 226, 46 (1997).

    Article  MATH  MathSciNet  Google Scholar 

  10. J. Ginibre, “Statistical ensembles of complex, quaternion, and real matrices,” J. Math. Phys. 6, 440 (1965).

    Article  MATH  MathSciNet  Google Scholar 

  11. V. L. Girko, “The circular law,” Theory Probab. Appl. 29, 694 (1985).

    Article  Google Scholar 

  12. V. L. Girko, Theory Probab. Appl. 30, 677 (1986).

    Article  MATH  MathSciNet  Google Scholar 

  13. F. Götze and A. N. Tikhomirov, On the Circular Law, Preprint arXiv: math/0702386v1 [math.PR] (2007).

    Google Scholar 

  14. F. Götze and A. N. Tikhomirov, “The circular law for random matrices,” Ann. Probab. 38, 1444 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  15. F. Götze and A. N. Tikhomirov, On the Asymptotic Spectrum of Products of Independent Random Matrices, Preprint. arXiv: 1012.2710.

  16. F. Hiai F. and D. Petz, “Asymptotic freeness almost everywhere for random matrices,” Acta. Sci. Math. (Szeged) 66, 809 (2000).

    MATH  MathSciNet  Google Scholar 

  17. R.A. Horn and C. R. Johnson, Matrix Analysis (Cambridge Univ. Press, Cambridge, 1990).

    MATH  Google Scholar 

  18. M. Krishnapur, “From random matrices to random analytic functions,” Ann. Probab. 37, 314 (2009).

    Article  MATH  MathSciNet  Google Scholar 

  19. V. A. Marchenko and L. A. Pastur, “Distribution of eigenvalues for some sets of random matrices,” Mat. Sb. 72(114), 507 (1967) [Math. USSR, Sb. 1, 457 (1967)].

    MathSciNet  Google Scholar 

  20. B. Maurey, “Some deviation inequalities,” Geom. Funct. Anal. 1, 188 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  21. A. Mays, A Real Quaternion Spherical Ensemble of Random Matrices, Preprint arXiv: 1209.0888 [mathph] (2012).

    Google Scholar 

  22. A. Naumov, Elliptic Law for Real Random Matrices, Preprint arXiv: 1201.1639 (2012).

    Google Scholar 

  23. L. R. Patricia, B. Biman, C. Charles, and A. R. Stuart, “Freezing of the classical two-dimensional, one-component plasma,” J. Chem. Phys. 81, 1406 (1984).

    Article  Google Scholar 

  24. T. Tao and V. Vu, “Random matrices: Universality of ESDs and the circular law (with an appendix by M. Krishnapur),” Ann. Probab. 38, 2023 (2010).

    Article  MATH  MathSciNet  Google Scholar 

  25. D. Voiculescu, “Limit laws for random matrices and free products,” Invent. Math. 104, 201 (1991).

    Article  MATH  MathSciNet  Google Scholar 

  26. D. Voiculescu, “Lectures on free probability theory,” Lectures on Probability Theory and Statistics (Saint-Flour, 1998), Lectures Notes in Math., N1738 (Springer, Berlin, 2000), 279.

    Google Scholar 

  27. E. Wigner, “Characteristic vectors of bordered matrices with infinite dimensions,” Ann. of Math. (2) 62, 548 (1955).

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Komi Scientific Center of UBRAS, Syktyvkar, 167000, Russia

    A. N. Tikhomirov

Authors
  1. A. N. Tikhomirov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. N. Tikhomirov.

Additional information

Original Russian Text © A. N. Tikhomirov, 2013, published in Matematicheskie Trudy, 2013, Vol. 16, No. 2, pp. 169–200.

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Tikhomirov, A.N. Asymptotic distribution of singular values for matrices in a spherical ensemble. Sib. Adv. Math. 24, 282–303 (2014). https://doi.org/10.3103/S105513441404004X

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.3103/S105513441404004X

Keywords

Search

Navigation