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The largest eigenvalue of small rank perturbations of Hermitian random matrices
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  • Published: 17 August 2005

The largest eigenvalue of small rank perturbations of Hermitian random matrices

  • S. Péché1 

Probability Theory and Related Fields volume 134, pages 127–173 (2006)Cite this article

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An Erratum to this article was published on 10 November 2005

Abstract

We compute the limiting eigenvalue statistics at the edge of the spectrum of large Hermitian random matrices perturbed by the addition of small rank deterministic matrices. We consider random Hermitian matrices with independent Gaussian entries M ij ,i≤j with various expectations. We prove that the largest eigenvalue of such random matrices exhibits, in the large N limit, various limiting distributions depending on both the eigenvalues of the matrix and its rank. This rank is also allowed to increase with N in some restricted way.

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References

  1. Baik, J., Ben Arous, G., Péché, S.: Phase transition of the largest eigenvalue for non-null complex sample covariance matrices. math.PR/0403022., 2004

  2. Bleher, P., Kuijlaars, B.: Large N limit of Gaussian random matrices with external source, part I. math-ph/0402042, 2004

  3. Brézin, E., Hikami, S.: Correlations of nearby levels induced by a random potential. Nucl. Phys. B 479, 697–706 (1996)

    MATH  Google Scholar 

  4. Deift, P., Kriecherbauer, T., McLaughlin, K., Venakides, S., Zhou, X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. Comm. Pure Appl. Math. 52, 1335–1425 (1999)

    MATH  MathSciNet  Google Scholar 

  5. Furedi, Z., Komlos, J.: The eigenvalues of random symmetric matrices. Combinatorica 1, 233–241 (1981)

    MathSciNet  Google Scholar 

  6. Geman, S.: A limit theorem for the norm of random matrices Ann. Prob. 8, 252–261 (1980)

    MATH  MathSciNet  Google Scholar 

  7. Harish-Chandra: Differential operators on a semisimple lie algebra. Am. J. Math. 79, 87–120 (1957)

    Google Scholar 

  8. Itzykson, C., Zuber, J.: The planar approximation. J. Math. Phys. 21, 411–421 (1957)

    MathSciNet  Google Scholar 

  9. Johansson, K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Comm. Math. Phys. 215, 683–705 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  10. Mehta, M.: Random matrices. Academic press, San Diego, second edition, 1991

  11. Péché, S.: Universality of local eigenvalue statistics for random sample covariance matrices. Ph.D. Thesis, Ecole Polytechnique Fédérale de Lausanne, 2003

  12. Soshnikov, A.: Universality at the edge of the spectrum in Wigner random matrices. Comm. Math. Phys. 207, 697–733 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  13. Tracy, C., Widom, H.: Level spacing distributions and the Airy kernel. Comm. Math. Phys. 159, 33–72 (1994)

    Article  Google Scholar 

  14. Wigner, E.: Characteristic vectors of bordered matrices with infinite dimensions. Ann. Math. 62, 548–564 (1955)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Institut Fourier, Université Joseph Fourier, BP 74, 38402, St MARTIN D'HERES Cedex, France

    S. Péché

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  1. S. Péché
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Correspondence to S. Péché.

Additional information

An erratum to this article is available at http://dx.doi.org/10.1007/s00440-005-0480-1.

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Péché, S. The largest eigenvalue of small rank perturbations of Hermitian random matrices. Probab. Theory Relat. Fields 134, 127–173 (2006). https://doi.org/10.1007/s00440-005-0466-z

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  • Received: 05 October 2004

  • Revised: 02 June 2005

  • Published: 17 August 2005

  • Issue Date: January 2006

  • DOI: https://doi.org/10.1007/s00440-005-0466-z

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Keywords

  • Stochastic Process
  • Probability Theory
  • Mathematical Biology
  • Random Matrice
  • Large Eigenvalue
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