Journal of Statistical Physics

, Volume 86, Issue 1, pp 109–147

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

  • L. Pastur
  • M. Shcherbina

DOI: 10.1007/BF02180200

Cite this article as:
Pastur, L. & Shcherbina, M. J Stat Phys (1997) 86: 109. doi:10.1007/BF02180200


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 matriceslocal asymptotic regimeuniversality conjectureorthogonal polynomial technique

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