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The Largest Eigenvalue of Real Symmetric, Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source, Part I

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Abstract

We consider the limiting location and limiting distribution of the largest eigenvalue in real symmetric (β=1), Hermitian (β=2), and Hermitian self-dual (β=4) random matrix models with rank 1 external source. They are analyzed in a uniform way by a contour integral representation of the joint probability density function of eigenvalues. Assuming the “one-band” condition and certain regularities of the potential function, we obtain the limiting location of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is not the critical value, and further obtain the limiting distribution of the largest eigenvalue when the nonzero eigenvalue of the external source matrix is greater than the critical value. When the nonzero eigenvalue of the external source matrix is less than or equal to the critical value, the limiting distribution of the largest eigenvalue will be analyzed in a subsequent paper. In this paper we also give a definition of the external source model for all β>0.

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References

  1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, vol. 55. For sale by the Superintendent of Documents, U.S. Government Printing Office, Washington, D.C. (1964)

    MATH  Google Scholar 

  2. Baik, J., Silverstein, J.W.: Eigenvalues of large sample covariance matrices of spiked population models. J. Multivar. Anal. 97(6), 1382–1408 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Baik, J., Wang, D.: On the largest eigenvalue of a Hermitian random matrix model with spiked external source I. Rank 1 case. Int. Math. Res. Not. 22, 5164–5240 (2011)

    Google Scholar 

  4. 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)

    Article  MathSciNet  MATH  Google Scholar 

  5. Bertola, M., Buckingham, R., Lee, S., Pierce, V.U.: Spectra of random Hermitian matrices with a small-rank external source: supercritical and subcritical regimes (2010). arXiv:1009.3894

  6. Bertola, M., Buckingham, R., Lee, S., Pierce, V.U.: Spectra of random Hermitian matrices with a small-rank external source: the critical and near-critical regimes (2010). arXiv:1011.4983

  7. Bleher, P., Delvaux, S., Kuijlaars, A.B.J.: Random matrices model with external source and a constrained vector equilibrium problem. Commun. Pure Appl. Math. 64(1), 116–160 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bloemendal, A., Virág, B.: Limits of spiked random matrices I (2010). arXiv:1011.1877

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

    Article  ADS  MATH  Google Scholar 

  10. Brézin, E., Hikami, S.: Level spacing of random matrices in an external source. Phys. Rev. E (3) 58(6, part A), 7176–7185 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  11. 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), 1–47 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  12. Deift, P., Gioev, D.: Universality at the edge of the spectrum for unitary, orthogonal, and symplectic ensembles of random matrices. Commun. Pure Appl. Math. 60(6), 867–910 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  13. Deift, P., Kriecherbauer, T., McLaughlin, K.T.-R.: New results on the equilibrium measure for logarithmic potentials in the presence of an external field. J. Approx. Theory 95(3), 388–475 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  14. 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(11), 1335–1425 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  15. Deift, P.A.: Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach. Courant Lecture Notes in Mathematics, vol. 3. New York University Courant Institute of Mathematical Sciences, New York (1999)

    Google Scholar 

  16. Dumitriu, I., Edelman, A., Shuman, G.: MOPS multivariate orthogonal polynomials (symbolically). J. Symb. Comput. 42(6), 587–620 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  17. Féral, D., Péché, S.: The largest eigenvalue of rank one deformation of large Wigner matrices. Comm. Math. Phys. 272 (2007)

  18. Forrester, P.J.: Log-gases Random Matrices. London Mathematical Society Monographs Series, vol. 34. Princeton University Press, Princeton (2010)

    MATH  Google Scholar 

  19. Forrester, P.J.: Probability densities and distributions for spiked Wishart β-ensembles (2011). arXiv:1101.2261

  20. Gross, K.I., Richards, D.S.P.: Special functions of matrix argument. I. Algebraic induction, zonal polynomials, and hypergeometric functions. Trans. Am. Math. Soc. 301(2), 781–811 (1987)

    MathSciNet  MATH  Google Scholar 

  21. Johansson, K.: On fluctuations of eigenvalues of random Hermitian matrices. Duke Math. J. 91(1), 151–204 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  22. Kriecherbauer, T., Shcherbina, M.: Fluctuations of eigenvalues of matrix models and their applications (2010). arXiv:1003.6121

  23. Macdonald, I.G.: Symmetric Functions and Hall Polynomials. 2nd edn. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (1995). With contributions by A. Zelevinsky, Oxford Science Publications

    MATH  Google Scholar 

  24. Mehta, M.L.: Random Matrices, 3rd edn. Pure and Applied Mathematics (Amsterdam), vol. 142. Elsevier/Academic Press, Amsterdam (2004)

    MATH  Google Scholar 

  25. Mo, M.Y.: The rank 1 real Wishart spiked model I. Finite N analysis (2010). arXiv:1011.5404

  26. Mo, M.Y.: The rank 1 real Wishart spiked model (2011). arXiv:1101.5144

  27. Muirhead, R.J.: Aspects of Multivariate Statistical Theory. Wiley Series in Probability and Mathematical Statistics. John Wiley & Sons Inc., New York (1982)

    Book  MATH  Google Scholar 

  28. Paul, D.: Asymptotics of sample eigenstructure for a large dimensional spiked covariance model. Stat. Sin. 17(4), 1617–1642 (2007)

    MATH  Google Scholar 

  29. Saff, E.B., Totik, V.: Logarithmic Potentials with External Fields. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 316. Springer, Berlin, (1997). Appendix B by Thomas Bloom

    MATH  Google Scholar 

  30. Shcherbina, M.: Edge universality for orthogonal ensembles of random matrices. J. Stat. Phys. 136(1), 35–50 (2009)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  31. Stanley, R.P.: Some combinatorial properties of Jack symmetric functions. Adv. Math. 77(1), 76–115 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  32. Wang, D.: The largest sample eigenvalue distribution in the rank 1 quaternionic spiked model of Wishart ensemble. Ann. Probab. 37(4), 1273–1328 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, University of Michigan, Ann Arbor, MI, 48105, USA

    Dong Wang

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  1. Dong Wang
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Correspondence to Dong Wang.

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Wang, D. The Largest Eigenvalue of Real Symmetric, Hermitian and Hermitian Self-dual Random Matrix Models with Rank One External Source, Part I. J Stat Phys 146, 719–761 (2012). https://doi.org/10.1007/s10955-012-0417-x

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  • DOI: https://doi.org/10.1007/s10955-012-0417-x

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