Ukrainian Mathematical Journal

, Volume 57, Issue 6, pp 936–966 | Cite as

A Simple Approach to the Global Regime of Gaussian Ensembles of Random Matrices

  • L. A. Pastur


We present simple proofs of several basic facts of the global regime (the existence and the form of the nonrandom limiting Normalized Counting Measure of eigenvalues, and the central limit theorem for the trace of the resolvent) for ensembles of random matrices whose probability law involves the Gaussian distribution. The main difference with previous proofs is the systematic use of the Poincare-Nash inequality, allowing us to obtain the O(n−2) bounds for the variance of the normalized trace of the resolvent that are valid up to the real axis in the spectral parameter.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    C. W. J. Beenakker, “Random-matrix theory of quantum transport,” Rev. Mod. Phys., 69, 731–847 (1997).Google Scholar
  2. 2.
    D. Bessis, C. Itzykson, and J.-B. Zuber, “Quantum field theory techniques in graphical enumeration,” Adv. Appl. Math., 1, 109–157 (1980).CrossRefMathSciNetGoogle Scholar
  3. 3.
    T. A. Brody, J. Flores, J. B. French, P. A. Mello, A. Pandey, and S. S. M. Wong, “Random-matrix physics: spectrum and strength fluctuations,” Rev. Mod. Phys., 53, 385–479 (1981).CrossRefMathSciNetGoogle Scholar
  4. 4.
    L. Pastur and M. Shcherbina, “Universality of the local eigenvalue statistics for a class of unitary invariant matrix ensembles,” J. Stat. Phys., 86, 109–147 (1997).CrossRefMathSciNetGoogle Scholar
  5. 5.
    K. Johansson, “On fluctuations of eigenvalues of random Hermitian matrices,” Duke Math. J., 91, 151–204 (1998).CrossRefMATHMathSciNetGoogle Scholar
  6. 6.
    V. L. Girko, Theory of Stochastic Canonical Equations, Kluwer, Dordrecht (2001).Google Scholar
  7. 7.
    A. Khorunzhy, B. Khoruzhenko, and L. Pastur, “Random matrices with independent entries: asymptotic properties of the Green function,” J. Math. Phys., 37, 5033–5060 (1996).CrossRefMathSciNetGoogle Scholar
  8. 8.
    L. H. Y. Chen, “An inequality involving normal distribution,” J. Multivar. Anal., 50, 13–223, 585-604 (1982).Google Scholar
  9. 9.
    M. Ledoux, Concentration of Measure Phenomenon, American Mathematical Society, Providence, RI (2001).Google Scholar
  10. 10.
    A. Khorunzhy, “Eigenvalue distribution of large random matrices with correlated entries,” Mat. Fiz. Anal. Geom., 3, 80–101 (1996).MATHMathSciNetGoogle Scholar
  11. 11.
    V. Marchenko and L. Pastur, “Eigenvalue distribution in some ensembles of random matrices,” Math. USSR Sbornik, 1, 457–483 (1967).Google Scholar
  12. 12.
    A. M. Tulino and S. Verdu, Random Matrix Theory and Wireless Communications, Now Publ. Inc., Hanover, MA (2004).Google Scholar
  13. 13.
    L. Pastur, “A simple approach to the global regime of the random matrix theory,” in: S. Miracle-Sole, J. Ruiz, and V. Zagrebnov (editors), Mathematical Results in Statistical Mechanics, World Scientific, Singapore (1999), pp. 429–454.Google Scholar
  14. 14.
    Z. D. Bai, “Methodologies in spectral analysis of large-dimensional random matrices: a review,” Statist. Sinica, 9, 611–677 (1999).MATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  • L. A. Pastur
    • 1
  1. 1.Mathematical Division of the Institute of Low-Temperature PhysicsKharkovUkraine

Personalised recommendations