Journal of Statistical Physics

, Volume 149, Issue 2, pp 246–258 | Cite as

On Eigenvalues of the Sum of Two Random Projections

  • V. Kargin


We study the behavior of eigenvalues of matrix P N +Q N where P N and Q N are two N-by-N random orthogonal projections. We relate the joint eigenvalue distribution of this matrix to the Jacobi matrix ensemble and establish the universal behavior of eigenvalues for large N. The limiting local behavior of eigenvalues is governed by the sine kernel in the bulk and by either the Bessel or the Airy kernel at the edge depending on parameters. We also study an exceptional case when the local behavior of eigenvalues of P N +Q N is not universal in the usual sense.


Random matrices Random projections Eigenvalues Universality Tracy–Widom distribution Sine kernel Airy kernel 


  1. 1.
    Baik, J., Borodin, A., Deift, P., Suidan, T.: A model for the bus system in Cuernavaca (Mexico). J. Phys., A, Math. Gen. 39, 8965–8975 (2006) MathSciNetADSMATHCrossRefGoogle Scholar
  2. 2.
    Bleher, P., Its, A.: Semiclassical asymptotics of orthogonal polynomials, Riemann–Hilbert problem, and universality in the matrix model. Ann. Math. 150, 185–266 (1999) MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Brezin, E., Hikami, S.: Extension of level-spacing universality. Phys. Rev. E 56, 264–269 (1997) MathSciNetADSCrossRefGoogle Scholar
  4. 4.
    Collins, B.: Product of random projections, Jacobi ensembles and universality problems arising from free probability. Probab. Theory Relat. Fields 133, 315–344 (2005) MATHCrossRefGoogle Scholar
  5. 5.
    Costin, O., Lebowitz, J.L.: Gaussian fluctuations in random matrices. Phys. Rev. Lett. 75, 69–72 (1995) ADSCrossRefGoogle Scholar
  6. 6.
    Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Commun. Pure Appl. Math. 52, 1335–1425 (1999) MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Erdos, L., Schlein, B., Yau, H.-T.: Universality of random matrices and local relaxation flow. Invent. Math. 185, 75–119 (2011) MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Erdos, L., Yau, H.-T.: Universality of local spectral statistics of random matrices. Bull. Am. Math. Soc. 49, 377–414 (2012). arXiv:1106.4986 MathSciNetCrossRefGoogle Scholar
  9. 9.
    Forrester, P.J.: The spectrum edge of random matrix ensembles. Nucl. Phys. B 402, 709–728 (1993) MathSciNetADSMATHCrossRefGoogle Scholar
  10. 10.
    Forrester, P.J.: Log-Gases and Random Matrices. Princeton University Press, Princeton (2010) MATHGoogle Scholar
  11. 11.
    Gustavsson, J.: Gaussian fluctuations of eigenvalues in the GUE. Ann. Inst. Henri Poincaré, B Probab. Stat. 41, 151–178 (2005) MathSciNetADSMATHCrossRefGoogle Scholar
  12. 12.
    James, A.T.: Distribution of matrix variates and latent roots derived from normal samples. Ann. Math. Stat. 35, 475–501 (1964) MATHCrossRefGoogle Scholar
  13. 13.
    Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215, 683–705 (2001) MathSciNetADSMATHCrossRefGoogle Scholar
  14. 14.
    Johnstone, I.M.: Multivariate analysis and Jacobi ensembles: largest eigenvalue, Tracy–Widom limits and rates of convergence. Ann. Stat. 36, 2638–2716 (2008) MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    Kuijlaars, A.B.J.: Universality. In: Akemann, G., Baik, J., Di Francesco, P. (eds.) The Oxford Handbook of Random Matrix Theory, pp. 103–134. Oxford University Press, Oxford (2011). arXiv:1103.5922 Google Scholar
  16. 16.
    Mehta, M.L.: Random Matrices, 3rd edn. Elsevier, Amsterdam (2004) MATHGoogle Scholar
  17. 17.
    Metcalfe, A.P.: Universality properties of Gelfand–Tsetlin patterns. Probab. Theory Relat. Fields (2011). doi: 10.1007/s00440-011-0399-7. Available at arXiv:1105.1272 Google Scholar
  18. 18.
    Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley, New York (1982) MATHCrossRefGoogle Scholar
  19. 19.
    Nagao, T., Forrester, P.J.: Asymptotic correlations at the spectrum edge of random matrices. Nucl. Phys. B 435, 401–420 (1995) MathSciNetADSMATHCrossRefGoogle Scholar
  20. 20.
    Pastur, L., Vasilchuk, V.: On the law of addition of random matrices. Commun. Math. Phys. 214, 249–286 (2000). arXiv:math-ph/0003043 MathSciNetADSMATHCrossRefGoogle Scholar
  21. 21.
    Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Commun. Math. Phys. 207, 697–733 (1999) MathSciNetADSMATHCrossRefGoogle Scholar
  22. 22.
    Soshnikov, A.: A note on the universality of the distribution of largest eigenvalues in certain sample covariance matrices. J. Stat. Phys. 108, 1033–1056 (2002) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Soshnikov, A.B.: Gaussian fluctuation for the number of particles in Airy, Bessel, sine, and other determinantal random point fields. J. Stat. Phys. 100, 491–522 (2000) MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Tao, T., Vu, V.: Random matrices: universality of the local eigenvalue statistics. Acta Math. 206, 127–204 (2011) MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Wachter, K.W.: The limiting empirical measure of multiple discriminant ratios. Ann. Stat. 8, 937–957 (1980) MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Statistical Laboratory, Department of MathematicsUniversity of CambridgeCambridgeUK

Personalised recommendations