Journal of Statistical Physics

, Volume 79, Issue 3, pp 585–611

On the statistical mechanics approach in the random matrix theory: Integrated density of states


  • A. Boutet de Monvel
    • Laboratory of Mathematical Physics and GeometryUniversité Paris VII
  • L. Pastur
    • Mathematical DivisionInstitute for Low Temperature Physics
  • M. Shcherbina
    • Mathematical DivisionInstitute for Low Temperature Physics

DOI: 10.1007/BF02184872

Cite this article as:
de Monvel, A.B., Pastur, L. & Shcherbina, M. J Stat Phys (1995) 79: 585. doi:10.1007/BF02184872


We consider the ensemble of random symmetricn×n matrices specified by an orthogonal invariant probability distribution. We treat this distribution as a Gibbs measure of a mean-field-type model. This allows us to show that the normalized eigenvalue counting function of this ensemble converges in probability to a nonrandom limit asn→∞ and that this limiting distribution is the solution of a certain self-consistent equation.

Key Words

Random matrixintegrating density of statesstatistical mechanicsmean field-theory

Copyright information

© Plenum Publishing Corporation 1995