Skip to main content
SpringerLink
Log in

The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices

  • Published:
Journal of Theoretical Probability Aims and scope Submit manuscript

Abstract

The probability that all eigenvalues of a product of m independent \(N \times N\) subblocks of a Haar distributed random real orthogonal matrix of size \((L_i+N) \times (L_i+N)\), \((i=1,\dots ,m)\) are real is calculated as a multidimensional integral, and as a determinant. Both involve Meijer G-functions. Evaluation formulae of the latter, based on a recursive scheme, allow it to be proved that for any m and with each \(L_i\) even the probability is a rational number. The formulae furthermore provide for explicit computation in small order cases.

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

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Adhikari, K., Reddy, N.K., Reddy, T.R., Saha, K.: Determinantal point processes in the plane from products of random matrices. Ann. Inst. H. Poincaré Probab. Stat. 52, 16–46 (2016)

    Article  MathSciNet  Google Scholar 

  2. Akemann, G., Kanzieper, E.: Integrable structure of Ginibre’s ensemble of real random matrices and a Pfaffian integration theorem. J. Stat. Phys. 129, 1159–1231 (2007)

    Article  MathSciNet  Google Scholar 

  3. Bergqvist, G., Forrester, P.J.: Rank probabilities for real random \(n \times n \times 2\) tensors. Elec. Commun. Probab. 16, 630–637 (2011)

    Article  MathSciNet  Google Scholar 

  4. de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. 19, 133–151 (1955)

    MathSciNet  MATH  Google Scholar 

  5. Edelman, A.: The probability that a random real Gaussian matrix has \(k\) real eigenvalues, related distributions, and the circular law. J. Multivar. Anal. 60, 203–232 (1997)

    Article  MathSciNet  Google Scholar 

  6. Fischmann, J.A.: Eigenvalue distributions on a single ring, Ph.D. thesis, Queen Mary, University of London (2012)

  7. Fischmann, J., Bruzda, W., Khoruzhenko, B.A., Sommers, H.-J., Zyczkowski, K.: Induced Ginibre ensemble of random matrices and quantum operations. J. Phys. A 45, 075203 (2012)

    Article  MathSciNet  Google Scholar 

  8. Forrester, P.J., Ipsen, J.R.: Log-gases and random matrices. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

  9. Forrester, P.J., Ipsen, J.R.: Probability of all eigenvalues real for products of standard Gaussian matrices. J. Phys. A 47, 065202 (2014)

    Article  MathSciNet  Google Scholar 

  10. Forrester, P.J., Ipsen, J.R.: Diffusion processes and the asymptotic bulk gap probability for the real Ginibre ensemble. J. Phys. A 48, 324001 (2015)

    Article  MathSciNet  Google Scholar 

  11. Forrester, P.J., Ipsen, J.R.: Real eigenvalue statistics for products of asymmetric real Gaussian matrices. Linear Algebra Appl. 510, 259–290 (2016)

    Article  MathSciNet  Google Scholar 

  12. Forrester, P.J., Mays, A.: Pfaffian point processes for the Gaussian real generalised eigenvalue problem. Prob. Theor. Rel. Field 154, 1–47 (2012)

    Article  Google Scholar 

  13. Forrester, P.J., Nagao, T.: Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99, 050603 (2007)

    Article  Google Scholar 

  14. Forrester, P.J., Nagao, T.: Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble. J. Phys. A 41, 375003 (2008)

    Article  MathSciNet  Google Scholar 

  15. Garcia del Molino, L.C., Pakdaman, K., Touboul, J.: Eigenvalues of non-symmetric random matrices: transitions and universality, arXiv:1605.00623, (2016)

  16. Garcia del Molino, L.C., Pakdaman, K., Touboul, J., Wainrib, G.: The real Ginibre ensemble with \(k = o(n)\) real eigenvalues. J. Stat. Phys. 162, 303–323 (2016)

    Article  MathSciNet  Google Scholar 

  17. Ipsen, J.R., Kieburg, M.: Weak commutation relations and eigenvalue statistics for products of rectangular random matrices. Phys. Rev. E 89, 032106 (2014)

    Article  Google Scholar 

  18. Kanzieper, E., Poplavskyi, M., Timm, C., Tribe, R., Zaboronski, O.: What is the probability that a large random matrix has no real eigenvalues? Ann. Appl. Probab. 65, 2733–2753 (2016)

    Article  MathSciNet  Google Scholar 

  19. Khoruzhenko, B.A., Sommers, H.-J., Zyczkowski, K.: Truncations of random orthogonal matrices. Phys. Rev. E 82, 040106 (2010)

    Article  MathSciNet  Google Scholar 

  20. Kumar, S.: Exact evaluations of some Meijer G-functions and probability of all eigenvalues real for products of two Gaussian matrices. J. Phys. A 48, 445206 (2015)

    Article  MathSciNet  Google Scholar 

  21. Lakshminarayan, A.: On the number of real eigenvalues of products of random matrices and an application to quantum entanglement. J. Phys. A 46, 152003 (2013)

    Article  MathSciNet  Google Scholar 

  22. Luke, Y.L.: The special functions and their approximations, vol. I. Academic Press, New York-London (1969)

    MATH  Google Scholar 

  23. Mays, A.: A geometrical triumvirate of real random matrices, Ph.D. thesis, University of Melbourne (2012)

  24. Simm, N.J.: Central limit theorems for the real eigenvalues of large random matrices. Random Matrice Theor. Appl. 06, 1750002 (2017)

    Article  MathSciNet  Google Scholar 

  25. ten Berge, J.B.: Kruskal’s polynomial for \(2 \times 2 \times 2\) arrays and a generalization to \(2 \times n \times n\) arrays. Psychometrika 56, 631–636 (1991)

    Article  Google Scholar 

  26. Wolfram Research Inc. Mathematica Version 10.0 (Wolfram Research Inc.: Champaign, Illinois)

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statistical Frontiers, The University of Melbourne, Victoria, 3010, Australia

    Peter J. Forrester

  2. Department of Physics, Shiv Nadar University, Gautam Buddha Nagar, Uttar Pradesh, 201314, India

    Santosh Kumar

Authors
  1. Peter J. Forrester
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Santosh Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Santosh Kumar.

Additional information

The work of PJF was supported by the Australian Research Council through Grant DP14102613.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Forrester, P.J., Kumar, S. The Probability That All Eigenvalues are Real for Products of Truncated Real Orthogonal Random Matrices. J Theor Probab 31, 2056–2071 (2018). https://doi.org/10.1007/s10959-017-0766-0

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10959-017-0766-0

Keywords

Mathematics Subject Classification (2010)

Search

Navigation