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On Universality for Orthogonal Ensembles of Random Matrices

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Abstract

We prove universality of local eigenvalue statistics in the bulk of the spectrum for orthogonal invariant matrix models with real analytic potentials with one interval limiting spectrum. Our starting point is the Tracy-Widom formula for the matrix reproducing kernel. The key idea of the proof is to represent the differentiation operator matrix written in the basis of orthogonal polynomials as a product of a positive Toeplitz matrix and a two diagonal skew symmetric Toeplitz matrix.

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References

  1. Albeverio S., Pastur L., Shcherbina M.: On Asymptotic Properties of the Jacobi Matrix Coefficients. Matem. Fiz. Analiz, Geom. 4, 263–277 (1997)

    MATH  MathSciNet  Google Scholar 

  2. Albeverio S., Pastur L., Shcherbina M.: On the 1/n expansion for some unitary invariant ensembles of random matrices. Commun. Math. Phys. 224, 271–305 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Boutetde Monvel A., Pastur L., Shcherbina M.: On the statistical mechanics approach in the random matrix theory. Integrated density of states. J. Stat. Phys. 79, 585–611 (1995)

    Article  ADS  MathSciNet  Google Scholar 

  4. Claeys T., Kuijalaars A.B.J.: Universality of the double scaling limit in random matrix models. Comm. Pure Appl. Math. 59, 1573–1603 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. 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, 388–475 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  6. 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. Commun. Pure Appl. Math. 52, 1335–1425 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Deift P., Kriecherbauer T., McLaughlin K., Venakides S., Zhou X.: Strong asymptotics of orthogonal polynomials with respect to exponential weights. Commun. Pure Appl. Math. 52, 1491–1552 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  8. Deift, P., Gioev, D.: Universality in random matrix theory for orthogonal and symplectic ensembles. Int. Math. Res. Papers. article ID rpm 004.116 pages (2007)

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

    Article  MATH  MathSciNet  Google Scholar 

  10. Deift P., Gioev D., Kriecherbauer T., Vanlessen M.: Universality for orthogonal and symplectic Laguerre-type ensembles. J.Stat.Phys 129, 949–1053 (2007)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Dyson D.J.: A Class of Matrix Ensembles. J.Math.Phys. 13, 90–107 (1972)

    Article  MATH  ADS  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  13. Mehta M.L.: Random Matrices. Academic Press, New York (1991)

    MATH  Google Scholar 

  14. Pastur L., Shcherbina M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stat. Phys. 86, 109–147 (1997)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  15. Pastur L., Shcherbina M.: On the edge universality of the local eigenvalue statistics of matrix models. Matem. Fiz. Analiz, Geom. 10(N3), 335–365 (2003)

    MATH  MathSciNet  Google Scholar 

  16. Pastur L., Shcherbina M.: Bulk universality and related properties of Hermitian matrix models. J. Stat. Phys. 130, 205–250 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  17. Reed M., Simon B.: Methods of Modern Mathematical Physics, vol. IV. Academic Press, New York (1978)

    Google Scholar 

  18. Saff E., Totik V.: Logarithmic Potentials with External Fields. Berlin:Springer-Verlag, (1997)

    MATH  Google Scholar 

  19. Shcherbina M.: Double scaling limit for matrix models with non analytic potentials. J. Math. Phys. 49, 033501–033535 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  20. Shcherbina, M.: On Universality for Orthogonal Ensembles of Random Matrices. http://arXiv.org/abs/math-ph/0701046, 2007

  21. Stojanovic A.: Universality in orthogonal and symplectic invariant matrix models with quatric potentials. Math. Phys. Anal. Geom. 3, 339–373 (2002)

    Article  MathSciNet  Google Scholar 

  22. Stojanovic, A.: Universalité pour des modéles orthogonale ou symplectiqua et a potentiel quartic. Math. Phys. Anal. Geom. Preprint Bibos 02-07-98; avaiable at http://www.physik.oni-bielefeld.de/bibos/preprints/02-07-08.pdf, 1988

  23. Tracy C.A., Widom H.: Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92, 809–835 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  24. Widom H.: On the relations between orthogonal, symplectic and unitary matrix models. J. Stat. Phys. 94, 347–363 (1999)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Mathematical Division of the Institute for Low Temperature Physics, Kharkov, Ukraine

    M. Shcherbina

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  1. M. Shcherbina
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Correspondence to M. Shcherbina.

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Communicated by A. Kupiainen

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Shcherbina, M. On Universality for Orthogonal Ensembles of Random Matrices. Commun. Math. Phys. 285, 957–974 (2009). https://doi.org/10.1007/s00220-008-0648-5

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