Journal of Statistical Physics

, Volume 86, Issue 1–2, pp 109–147

Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles

  • L. Pastur
  • M. Shcherbina


This paper is devoted to the rigorous proof of the universality conjecture of random matrix theory, according to which the limiting eigenvalue statistics ofn×n random matrices within spectral intervals ofO(n−1) is determined by the type of matrix (real symmetric, Hermitian, or quaternion real) and by the density of states. We prove this conjecture for a certain class of the Hermitian matrix ensembles that arise in the quantum field theory and have the unitary invariant distribution defined by a certain function (the potential in the quantum field theory) satisfying some regularity conditions.

Key Words

Random matrices local asymptotic regime universality conjecture orthogonal polynomial technique 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    M. L. Mehta,Random Matrices (Academic Press, New York, 1991).Google Scholar
  2. 2.
    D. Fox and P. Kahn,Phys. Rev. 134:B1151 (1964).Google Scholar
  3. 3.
    H. Leff,J. Math. Phys. 5:761 (1964).Google Scholar
  4. 4.
    F. J. Dyson,J. Math. Phys. 13:90 (1972).Google Scholar
  5. 5.
    D. Bessis, C. Itzykson, and J. Zuber,Adv. Appl. Math. 1:109 (1980).Google Scholar
  6. 6.
    K. Demetrefi,Int. J. Mod. Phys. A 8:1185 (1993).Google Scholar
  7. 7.
    J.-L. Pichard, InQuantum Coherence in Mesoscopic Systems, B. Kramer, ed. (Plenum Press, New York, 1991).Google Scholar
  8. 8.
    R. Fernandez, J. Frohlich, and A. Sokal,Random Walks, Critical Phenomena and Triviality in the Quantum Field Theory (Springer-Verlag, Berlin, 1992).Google Scholar
  9. 9.
    A. Boutet de Monvel, L. Pastur, and M. Shcherbina,J. Stat. Phys. 79:585 (1995).Google Scholar
  10. 10.
    G. Szego,Orthogonal Polynomials (AMS, New York, 1959).Google Scholar
  11. 11.
    D. Lubinsky,Strong Asymptotics for Extremal Polynomials Associated with the Erdos-type Weights (Longmans, Harlow, 1989).Google Scholar
  12. 12.
    E. Rakhmanov,Lectures Notes in Mathematics, No. 1150 (Springer, Berlin, 1993).Google Scholar
  13. 13.
    L. Pastur,Lett. Math. Phys. 25:259 (1992).Google Scholar
  14. 14.
    G. Moore,Prog. Theor. Phys. Suppl. 102:225 (1990).Google Scholar
  15. 15.
    E. Brezin and A. Zee,Nucl. Phys. B 402:613 (1993).Google Scholar
  16. 16.
    B. Eynard, Large random matrices: Eigenvalue distribution, Preprint SPhT/93-999, Saclay (1993).Google Scholar
  17. 17.
    G. Hackenbroich and H. A. Weidenmuller, University of random matrix results for non Gaussian ensembles, Preprint, Heidelberg University (1994).Google Scholar
  18. 18.
    C. Tracy and H. Widom,Commun. Math. Phys. 163:35 (1994).Google Scholar

Copyright information

© Plenum Publishing Corporation 1997

Authors and Affiliations

  • L. Pastur
    • 1
    • 2
  • M. Shcherbina
    • 1
  1. 1.Mathematical Division of the Institute for Low Temperature Physics of the NationalAcademy of Sciences of UkraineKharkovUkraine
  2. 2.U.F.R. de MathématiquesUniversité Paris VIIParisFrance

Personalised recommendations