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Journal of Theoretical Probability

, Volume 30, Issue 4, pp 1624–1654 | Cite as

Complex Outliers of Hermitian Random Matrices

  • Jean Rochet
Article

Abstract

In this paper, we study the asymptotic behavior of the outliers of the sum a Hermitian random matrix and a finite rank matrix which is not necessarily Hermitian. We observe several possible convergence rates and outliers locating around their limits at the vertices of regular polygons as in Benaych-Georges and Rochet (Probab Theory Relat Fields, 2015), as well as possible correlations between outliers at macroscopic distance as in Knowles and Yin (Ann Probab 42(5):1980–2031, 2014) and Benaych-Georges and Rochet (2015). We also observe that a single spike can generate several outliers in the spectrum of the deformed model, as already noticed in Benaych-Georges and Nadakuditi (Adv Math 227(1):494–521, 2011) and Belinschi et al. (Outliers in the spectrum of large deformed unitarily invariant models 2012, arXiv:1207.5443v1). In the particular case where the perturbation matrix is Hermitian, our results complete the work of Benaych-Georges et al. (Electron J Probab 16(60):1621–1662, 2011), as we consider fluctuations of outliers lying in “holes” of the limit support, which happen to exhibit surprising correlations.

Keywords

Random matrices Spiked models Extreme eigenvalue statistics Gaussian fluctuations 

Mathematics Subject Classification

15B52 60F05 

References

  1. 1.
    Anderson, G., Guionnet, A., Zeitouni, O.: An Introduction to Random Matrices. Cambridge Studies in Advanced Mathematics, vol. 118. Cambridge University Press (2009)Google Scholar
  2. 2.
    Bai, Z.D., Silverstein, J.W.: Spectral Analysis of Large Dimensional Random Matrices, 2nd edn. Springer, NewYork (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)CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Beardon, A.: Complex Analysis: The Winding Number Principle in Analysis and Topology. Wiley, NewYork (1979)zbMATHGoogle Scholar
  5. 5.
    Belinschi, S.T., Bercovici, H., Capitaine, M., Février, M.: Outliers in the spectrum of large deformed unitarily invariant models. arXiv:1207.5443v1, (2012)
  6. 6.
    Benaych-Georges, F., Guionnet, A., Maida, M.: Fluctuations of the extreme eigenvalues of finite rank deformations of random matrices. Electron. J. Probab. 16(60), 1621–1662 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Benaych-Georges, F., Guionnet, A., Maida, M.: Large deviations of the extreme eigenvalues of random deformations of matrices. Probab. Theory Relat. Fields 154(3), 703–751 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Benaych-Georges, F., Nadakuditi, R.R.: The eigenvalues and eigenvectors of finite, low rank perturbations of large random matrices. Adv. Math. 227(1), 494–521 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Benaych-Georges, F., Nadakuditi, R.R.: The singular values and vectors of low rank perturbations of large rectangular random matrices. J. Multivar. Anal. 111, 120–135 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Benaych-Georges, F., Rochet, J.: Outliers in the single ring theorem. Probab. Theory Relat. Fields (2015). doi: 10.1007/s00440-015-0632-x
  11. 11.
    Benaych-Georges, F., Rochet, J.: Fluctuations for analytic test functions in the single ring theorem. arXiv preprint arXiv:1504.05106, (2015)
  12. 12.
    Bordenave, C., Capitaine, M.: Outlier eigenvalues for deformed i.i.d. random matrices. arXiv:1403.6001
  13. 13.
    Capitaine, M., Donati-Martin, C., Féral, D.: The largest eigenvalue of finite rank deformation of large Wigner matrices: convergence and non universality of the fluctuations. Ann. Probab. 37(1), 1–47 (2009)CrossRefzbMATHMathSciNetGoogle Scholar
  14. 14.
    Capitaine, M., Donati-Martin, C., Féral, D.: Central limit theorems for eigenvalues of deformations of Wigner matrices. Ann. I.H.P. Probab. Stat. 48(1), 107–133 (2012)zbMATHMathSciNetGoogle 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. Probab. 16(64), 1750–1792 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  16. 16.
    Erdös, L., Yau, H.T., Yin, J.: Bulk universality for generalized Wigner matrices. arXiv preprint arXiv:1001.3453
  17. 17.
    Féral, D., Péché, S.: The largest eigenvalue of rank one deformation of large Wigner matrices. Commun. Math. Phys. 272, 185–228 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Fyodorov, Y.V., Sommers, H.-J.: Statistics of S-matrix poles in few-channel chaotic scattering: crossover from isolated to overlapping resonances. JETP Lett. 63, 1026–1030 (1996)CrossRefGoogle Scholar
  19. 19.
    Fyodorov, Y.V., Sommers, H.-J.: Statistics of resonance poles, phase shifts and time delays in quantum chaotic scattering: random matrix approach for systems with broken time reversal invariance. J. Math. Phys. 38, 1918–1981 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Fyodorov, Y.V., Khoruzhenko, B.A.: Systematic analytical approach to correlation functions of resonances in quantum chaotic scattering. Phys. Rev. Lett. 83, 65–68 (1999)CrossRefGoogle Scholar
  21. 21.
    Fyodorov, Y.V., Sommers, H.-J.: Random matrices close to Hermitian or unitary: overview of methods and results. J. Phys. A Math. Gen. 36, 3303–3347 (2003)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Horn, R.A., Johnson, C.R.: Matrix Analysis, Cambridge University Press, ISBN 978-0-521-38632-6 (1985)Google Scholar
  23. 23.
    Johnstone, I.M.: On the distribution of the largest eigenvalue in principal components analysis. Ann. Stat. 29(2), 295–327 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Knowles, A., Yin, J.: The isotropic semicircle law and deformation of Wigner matrices. Commun. Pure Appl. Math. 66(11), 1663–1750 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Knowles, A., Yin, J.: The outliers of a deformed Wigner matrix. Ann. Probab. 42(5), 1980–2031 (2014)CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Mingo, J.A., Śniady, P., Speicher, R.: Second order freeness and fluctuations of random matrices: II. Unitary random matrices. Adv. Math. 209(1), 212–240 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Rajagopalan, A.B.: Outlier eigenvalue fluctuations of perturbed iid matrices. arXiv preprint arXiv:1507.01441
  28. 28.
    O’Rourke, S., Renfrew, D.: Low rank perturbations of large elliptic random matrices, arXivGoogle Scholar
  29. 29.
    Péché, S.: The largest eigenvalue of small rank perturbations of Hermitian random matrices. Prob. Theory Relat. Fields 134, 127–173 (2006)CrossRefzbMATHMathSciNetGoogle Scholar
  30. 30.
    Pizzo, A., Renfrew, D., Soshnikov, A.: On finite rank deformations of Wigner matrices. Ann. Inst. H. Poincaré Probab. Stat. 49(1), 64–94 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  31. 31.
    Pizzo, A., Renfrew, D., Soshnikov, A.: Fluctuations of matrix entries of regular functions of Wigner matrices. J. Stat. Phys. 146(3), 550–591 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  32. 32.
    Tao, T.: Outliers in the spectrum of iid matrices with bounded rank perturbations. Probab. Theory Relat. Fields 155, 231–263 (2013)CrossRefzbMATHMathSciNetGoogle Scholar
  33. 33.
    Sinai, Y., Soshnikov, A.: Central limit theorem for traces of large random symmetric matrics with independent matrix elements. Bull. Braz. Math. Soc. 29(1), 1–24 (1998)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

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

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