Skip to main content
SpringerLink
Log in

Limit Correlation Functions for Fixed Trace Random Matrix Ensembles

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Universal limits for the eigenvalue correlation functions in the bulk of the spectrum are shown for a class of nondeterminantal random matrices known as the fixed trace or the Hilbert-Schmidt ensemble. These universal limits have been proved before for determinantal Hermitian matrix ensembles and for some special classes of the Wigner random matrices.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Akemann G., Cicuta G.M., Molinari L., Vernizzi G.: Compact support probability distributions in random matrix theory. Phys. Rev. E (3) 59(2, part A), 1489–1497 (1999)

    Article  ADS  MathSciNet  Google Scholar 

  2. Akemann G., Vernizzi G.: Macroscopic and microscopic (non-)universality of compact support random matrix theory. Nucl. Phys. B 583(3), 739–757 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  3. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X.: Uniform asymptotics for polynomials orthogonal with respect to varying exponential weights and applications to universality questions in random matrix theory. J. Comm. Pure Appl. Math. 52(11), 1335–1425 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  4. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X.: Strong asymptotics for polynomials orthogonal with respect to exponential weights. J. Comm. Pure Appl. Math. 52(12), 1491–1552 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  5. Deift, P. A.: Orthogonal polynomials and random matrices: a Riemann-Hilbert approach. Courant Lecture Notes in Mathematics, Vol. 3. N. Y.: New York University/Courant Institute of Mathematical Sciences, 1999

  6. Deift P., Kriecherbauer T., McLaughlin K.T.-R., Venakides S., Zhou X.: A Riemann-Hilbert approach to asymptotic questions for orthogonal polynomials. J. Comput. Appl. Math. 133(1–2), 47–63 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  7. Feller, W.: An introduction to probability theory and its applications. Vol. II. Second edition. New York: John Wiley & Sons Inc., 1971

  8. Forrester, P.J.: Log-gases and random matrices. Book in progress. http://www.ms.unimelb.edu.au/~matpjf/matpjf.html, 2005

  9. Götze, F., Gordin, M., Levina, A.: Limit correlation function at zero for fixed trace random matrix ensembles.(Russian) Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 341, 68–80 (2007); translation in J. Math. Sci. (N.Y.) 147(4), 6884–6890 (2007)

  10. Johansson K.: Universality of the local spacing distribution in certain ensembles of Hermitian Wigner matrices. Commun. Math. Phys. 215(3), 683–705 (2001)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  11. Mehta, M. L.: Random matrices. Third edition. Pure and Applied Mathematics (Amsterdam), 142. Amsterdam: Elsevier/Academic Press, 2004

  12. Pastur L., Shcherbina M.: Universality of the local eigenvalue statistics for a class of unitary invariant random matrix ensembles. J. Stati. Phys. 86(1–2), 109–147 (1997)

    Article  MATH  MathSciNet  ADS  Google Scholar 

  13. Petrov, V.V.: Sums of independent random variables. N.Y.: Springer, 1975

  14. Rosenzweig, N.: Statistical mechanics of equally likely quantum systems. In: Statistical physics (Brandeis Summer Institute, 1962, Vol. 3), N. Y.: W. A. Benjamin, (1963), pp. 91–158

  15. Soshnikov A.: Determinantal point random fields. Russ. Math. Surv. 55(5), 923–975 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  16. Szego, G.: Orthogonal polynomials. Fourth edition. Providence, R.I.: Amer. Math. Soc., 1975

  17. Tracy C.A., Widom H.: Correlation functions, cluster functions, and spacing distributions for random matrices. J. Stat. Phys. 92(5–6), 809–835 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  18. Van Assche W., Geronimo J.S.: Asymptotics for orthogonal polynomials with regularly varying recurrence coefficients. Rocky Mountain J. Math. 19(1), 39–49 (1989)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Bielefeld, Postfach 100131, D-33501, Bielefeld, Germany

    Friedrich Götze

  2. V.A. Steklov Mathematical Institute at St. Petersburg, 27 Fontanka emb., St. Petersburg, 191023, Russia

    Mikhail Gordin

Authors
  1. Friedrich Götze
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mikhail Gordin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Friedrich Götze.

Additional information

Communicated by H. Spohn

Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik”.

Research supported by Sonderforschungsbereich 701 “Spektrale Strukturen und Topologische Methoden in der Mathematik,” and grants RFBR-05-01-00911, DFG-RFBR-04-01-04000, and NS-638.2008.1.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Götze, F., Gordin, M. Limit Correlation Functions for Fixed Trace Random Matrix Ensembles. Commun. Math. Phys. 281, 203–229 (2008). https://doi.org/10.1007/s00220-008-0484-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0484-7

Keywords

Search

Navigation