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Journal of Theoretical Probability

, Volume 31, Issue 4, pp 2056–2071 | Cite as

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

  • Peter J. Forrester
  • Santosh KumarEmail author
Article

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.

Keywords

Random matrix products Truncated orthogonal matrices Probability of real eigenvalues Meijer G-functions Arithmetic structures 

Mathematics Subject Classification (2010)

15A52 15A15 15A18 33E20 11B37 

References

  1. 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)MathSciNetCrossRefGoogle Scholar
  2. 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)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Bergqvist, G., Forrester, P.J.: Rank probabilities for real random \(n \times n \times 2\) tensors. Elec. Commun. Probab. 16, 630–637 (2011)MathSciNetCrossRefGoogle Scholar
  4. 4.
    de Bruijn, N.G.: On some multiple integrals involving determinants. J. Indian Math. Soc. 19, 133–151 (1955)MathSciNetzbMATHGoogle Scholar
  5. 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)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Fischmann, J.A.: Eigenvalue distributions on a single ring, Ph.D. thesis, Queen Mary, University of London (2012)Google Scholar
  7. 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)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Forrester, P.J., Ipsen, J.R.: Log-gases and random matrices. Princeton University Press, Princeton (2010)zbMATHGoogle Scholar
  9. 9.
    Forrester, P.J., Ipsen, J.R.: Probability of all eigenvalues real for products of standard Gaussian matrices. J. Phys. A 47, 065202 (2014)MathSciNetCrossRefGoogle Scholar
  10. 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)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Forrester, P.J., Ipsen, J.R.: Real eigenvalue statistics for products of asymmetric real Gaussian matrices. Linear Algebra Appl. 510, 259–290 (2016)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Forrester, P.J., Mays, A.: Pfaffian point processes for the Gaussian real generalised eigenvalue problem. Prob. Theor. Rel. Field 154, 1–47 (2012)CrossRefGoogle Scholar
  13. 13.
    Forrester, P.J., Nagao, T.: Eigenvalue statistics of the real Ginibre ensemble. Phys. Rev. Lett. 99, 050603 (2007)CrossRefGoogle Scholar
  14. 14.
    Forrester, P.J., Nagao, T.: Skew orthogonal polynomials and the partly symmetric real Ginibre ensemble. J. Phys. A 41, 375003 (2008)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Garcia del Molino, L.C., Pakdaman, K., Touboul, J.: Eigenvalues of non-symmetric random matrices: transitions and universality, arXiv:1605.00623, (2016)
  16. 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)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Ipsen, J.R., Kieburg, M.: Weak commutation relations and eigenvalue statistics for products of rectangular random matrices. Phys. Rev. E 89, 032106 (2014)CrossRefGoogle Scholar
  18. 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)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Khoruzhenko, B.A., Sommers, H.-J., Zyczkowski, K.: Truncations of random orthogonal matrices. Phys. Rev. E 82, 040106 (2010)MathSciNetCrossRefGoogle Scholar
  20. 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)MathSciNetCrossRefGoogle Scholar
  21. 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)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Luke, Y.L.: The special functions and their approximations, vol. I. Academic Press, New York-London (1969)zbMATHGoogle Scholar
  23. 23.
    Mays, A.: A geometrical triumvirate of real random matrices, Ph.D. thesis, University of Melbourne (2012)Google Scholar
  24. 24.
    Simm, N.J.: Central limit theorems for the real eigenvalues of large random matrices. Random Matrice Theor. Appl. 06, 1750002 (2017)MathSciNetCrossRefGoogle Scholar
  25. 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)CrossRefGoogle Scholar
  26. 26.
    Wolfram Research Inc. Mathematica Version 10.0 (Wolfram Research Inc.: Champaign, Illinois)Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.School of Mathematics and Statistics, ARC Centre of Excellence for Mathematical and Statistical FrontiersThe University of MelbourneVictoriaAustralia
  2. 2.Department of PhysicsShiv Nadar UniversityGautam Buddha NagarIndia

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