Probability Theory and Related Fields

, Volume 165, Issue 1–2, pp 313–363 | Cite as

Outliers in the Single Ring Theorem

  • Florent Benaych-Georges
  • Jean Rochet


This text is about spiked models of non-Hermitian random matrices. More specifically, we consider matrices of the type \({\mathbf {A}}+{\mathbf {P}}\), where the rank of \({\mathbf {P}}\) stays bounded as the dimension goes to infinity and where the matrix \({\mathbf {A}}\) is a non-Hermitian random matrix, satisfying an isotropy hypothesis: its distribution is invariant under the left and right actions of the unitary group. The macroscopic eigenvalue distribution of such matrices is governed by the so called Single Ring Theorem, due to Guionnet, Krishnapur and Zeitouni. We first prove that if \({\mathbf {P}}\) has some eigenvalues out of the maximal circle of the single ring, then \({\mathbf {A}}+{\mathbf {P}}\) has some eigenvalues (called outliers) in the neighborhood of those of \({\mathbf {P}}\), which is not the case for the eigenvalues of \({\mathbf {P}}\) in the inner cycle of the single ring. Then, we study the fluctuations of the outliers of \({\mathbf {A}}\) around the eigenvalues of \({\mathbf {P}}\) and prove that they are distributed as the eigenvalues of some finite dimensional random matrices. Such kind of fluctuations had already been shown for Hermitian models. More surprising facts are that outliers can here have very various rates of convergence to their limits (depending on the Jordan Canonical Form of \({\mathbf {P}}\)) and that some correlations can appear between outliers at a macroscopic distance from each other (a fact already noticed by Knowles and Yin in (Ann Probab 42:1980–2031, 2014) in the Hermitian case, but only for non Gaussian models, whereas spiked Gaussian matrices belong to our model and can have such correlated outliers). Our first result generalizes a result by Tao proved specifically for matrices with i.i.d. entries, whereas the second one (about the fluctuations) is new.


Random matrices Spiked models Extreme eigenvalue statistics Gaussian fluctuations Ginibre matrices 

Mathematics Subject Classification

15A52 60F05 



We would like to thank J. Novak for discussions on Weingarten calculus.


  1. 1.
    Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge studies in advanced mathematics, 118 (2009)Google Scholar
  2. 2.
    Bai, Z.D., Silverstein, J.W.: Spectral analysis of large dimensional random matrices, 2nd edn. Springer, New York (2009)zbMATHGoogle Scholar
  3. 3.
    Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for nonnull complex sample covariance matrices. Ann. Probab. 33(5), 1643–1697 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Basak, A., Dembo, A.: Limiting spectral distribution of sums of unitary and orthogonal matrices. Electron. Commun. Probab. 18(69), 19 (2013)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Beardon, A.: Complex Analysis: the Winding Number principle in analysis and topology. Wiley, New York (1979)zbMATHGoogle Scholar
  6. 6.
    Benaych-Georges, F.: Exponential bounds for the support convergence in the Single Ring Theorem. J. Funct. Anal. 268, 3492–3507 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Benaych-Georges, F., Guionnet, A., Maida M.: Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Electron. J. Prob. 16 (2011) (Paper no. 60, 1621–1662)Google Scholar
  8. 8.
    Benaych-Georges, F., Guionnet, A., Maida, M.: Large deviations of the extreme eigenvalues of random deformations of matrices. Probab. Theory Related Fields 154(3), 703–751 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Benaych-Georges, F., Rao, R.N.: The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Adv. Math. 227(1), 494–521 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Benaych-Georges, F., Rao, R.N.: The singular values and vectors of low rank perturbations of large rectangular random matrices. J. Multivariate Anal. 111, 120–135 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Bordenave, C., Capitaine, M.: Outlier eigenvalues for deformed i.i.d. random matrices. arXiv:1403.6001
  12. 12.
    Bordenave, C., Chafaï, D.: Around the circular law. Probab. Surv. 9, 1–89 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Capitaine, M., Donati-Martin, C., Féral, D.: The largest eigenvalues of finite rank deformation of large Wigner matrices: convergence and nonuniversality of the fluctuations. Ann. Probab. 37, 1–47 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Capitaine, M., Donati-Martin, C., Féral, D.: Central limit theorems for eigenvalues of deformations of Wigner matrices. Ann. Inst. Henri Poincaré Probab. Stat. 48(1), 107–133 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Capitaine, M., Donati-Martin, C., Féral, D., Février, M.: Free convolution with a semi-circular distribution and eigenvalues of spiked deformations of Wigner matrices. Electron. J. Prob. 16, 1750–1792 (2011)CrossRefzbMATHGoogle Scholar
  16. 16.
    Collins, B., Śniady, P.: Integration with respect to the Haar measure on unitary, orthogonal and symplectic group. Comm. Math. Phys. 264(3), 773–795 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Féral, D., Péché, S.: The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys. 272, 185–228 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Guionnet, A., Krishnapur, M., Zeitouni, O.: The Single Ring Theorem. Ann. Math. 174(2), 1189–1217 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Guionnet, A., Zeitouni, O.: Support convergence in the Single Ring Theorem. Probab. Theory Related Fields 154(3–4), 661–675 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Hiai, F., Petz D.: The semicircle law, free random variables, and entropy. Amer. Math. Soc., Math. Surv. Monogr. 77 (2000)Google Scholar
  21. 21.
    Horn, R.A., Johnson, C.R.: Matrix Analysis. Cambridge University Press. ISBN: 978-0-521-38632-6 (1985)Google Scholar
  22. 22.
    Jiang, T.: Maxima of entries of Haar distributed matrices. Probab. Theory Related Fields 131(1), 121–144 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Kallenberg, O.: Foundations of Modern Probability. Springer, New York (1997)zbMATHGoogle Scholar
  24. 24.
    Knowles, A., Yin, J.: The isotropic semicircle law and deformation of Wigner matrices. Comm. Pure Appl. Math. 66(11), 1663–1750 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Knowles, A., Yin, J.: The outliers of a deformed Wigner matrix. Ann. Probab. 42(5), 1980–2031 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    O’Rourke, S., Renfrew, D.: Low rank perturbations of large elliptic random matrices. Electron. J. Probab. 19(43), 65 (2014)MathSciNetzbMATHGoogle Scholar
  27. 27.
    Péché, S.: The largest eigenvalue of small rank perturbations of Hermitian random matrices. Prob. Theory Relat. Fields 134, 127–173 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Rudelson, M., Vershynin, R.: Invertibility of random matrices: unitary and orthogonal perturbations. J. Amer. Math. Soc. 27, 293–338 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Tao, T.: Topics in random matrix theory, Graduate Studies in Mathematics, AMS (2012)Google Scholar
  30. 30.
    Tao, T.: Outliers in the spectrum of i.i.d. matrices with bounded rank perturbations. Probab. Theory Related Fields 155(1–2), 231–263 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Zhang, C., Qiu, R.C.: Data Modeling with Large Random Matrices in a Cognitive Radio Network Testbed: initial experimental demonstrations with 70 Nodes. arXiv:1404.3788

Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.MAP5, Université Paris DescartesParis Cedex 06France

Personalised recommendations