Mathematical Physics, Analysis and Geometry

, Volume 3, Issue 4, pp 339–373

Universality in Orthogonal and Symplectic Invariant Matrix Models with Quartic Potential

  • Alexandre Stojanovic
Article

Abstract

In this work, we develop an orthogonal-polynomials approach for random matrices with orthogonal or symplectic invariant laws, called one-matrix models with polynomial potential in theoretical physics, which are a generalization of Gaussian random matrices. The representation of the correlation functions in these matrix models, via the technique of quaternion determinants, makes use of matrix kernels. We get new formulas for matrix kernels, generalizing the known formulas for Gaussian random matrices, which essentially express them in terms of the reproducing kernel of the theory of orthogonal polynomials. Finally, these formulas allow us to prove the universality of the local statistics of eigenvalues, both in the bulk and at the edge of the spectrum, for matrix models with two-band quartic potential by using the asymptotics given by Bleher and Its for the corresponding orthogonal polynomials.

random matrices eigenvalues correlation function universality conjecture orthogonal polynomials 

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References

  1. 1.
    Abramowitz, M. and Stegun, I. A. (eds): Handbook of Mathematical Functions, Dover, New York, 1968.Google Scholar
  2. 2.
    Bleher, P. and Its, A.: Semi-classical asymptotics of orthogonal polynomials, Riemann-Hilbert problem and universality in the matrix model, Ann. of Math. 150(1999), 185–266.Google Scholar
  3. 3.
    Bleistein, N. and Handelsman, R. A.: Asymptotic Expansions of Integrals, Dover, New York, 1986.Google Scholar
  4. 4.
    Boutet de Monvel, A., Pastur, L. and Shcherbina, M.: On the statistical mechanics approach to the random matrix theory: the integrated density of states, J. Statist. Phys. 79(1995), 585–611.Google Scholar
  5. 5.
    Deift, P., Kriecherbauer, T., McLaughlin, K. T.-R., Venakides, S. and Zhou, X.: Asymptotics for polynomial orthogonal with respect to varying exponential weight, Internat. Math. Res. Notes 16(1997), 759–782.Google Scholar
  6. 6.
    DiFrancesco, P., Ginsparg, P. and Zinn-Justin, J.: 2D gravity and random matrices, Phys. Rep. 254(1995), 1–133.Google Scholar
  7. 7.
    Forrester, P. J., Nagao, T. and Honner, G.: Correlations for the orthogonal-unitary and symplectic-unitary transitions at the hard and soft edges, Nuclear Phys. B 553[PM] (1999), 601–643.Google Scholar
  8. 8.
    Johansson, K.: On fluctuation of eigenvalues of random Hermitian matrices, Duke Math. J. 91(1998), 151–204.Google Scholar
  9. 9.
    Mehta, M. L.: Random Matrices, and the Statistical Theory of Energy Levels, Academic Press, New York, 1967.Google Scholar
  10. 10.
    Mehta, M. L.: Matrix Theory, Selected Topics and Useful Results, Les Éditions de physique, France, 1989.Google Scholar
  11. 11.
    Mehta, M. L.: Random Matrices, Academic Press, New York, 1991.Google Scholar
  12. 12.
    Pastur, L. and Shcherbina, M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles, J. Statist. Phys. 86(1997), 109–147.Google Scholar
  13. 13.
    Stojanovic, A.: Une approche par les polynômes orthogonaux pour des classes de matrices aléatoires orthogonalement et symplectiquement invariantes: application à l'universalité de la statistique locale des valeurs propres, Preprint, www.physik.uni-bielefeld.de/bibos/preprints, 00–01–06.Google Scholar
  14. 14.
    Szego, G.: Orthogonal Polynomials, Amer. Math. Soc., Providence, 1939.Google Scholar
  15. 15.
    Tracy, C. A. and Widom, H.: Orthogonal and symplectic matrix ensembles, Comm. Math. Phys. 177(1996), 727–754.Google Scholar
  16. 16.
    Tracy, C. A. and Widom, H.: Correlation functions, cluster functions and spacing distributions for random matrices, J. Statist. Phys. 92(1998), 809–835.Google Scholar
  17. 17.
    Widom, H.: On the relation between orthogonal, symplectic and unitary matrix ensembles, J. Statist. Phys. 94(1999), 347–364.Google Scholar

Copyright information

© Kluwer Academic Publishers 2000

Authors and Affiliations

  • Alexandre Stojanovic
    • 1
  1. 1.Institut de Mathématiques, Physique-mathématique et géométrieUniversité Paris 7Paris cedex 05France

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