Skip to main content
SpringerLink
Log in

Supersymmetry Approach to Wishart Correlation Matrices: Exact Results

  • Published:
Journal of Statistical Physics Aims and scope Submit manuscript

Abstract

We calculate the ‘one-point function’, meaning the marginal probability density function for any single eigenvalue, of real and complex Wishart correlation matrices. No explicit expression had been obtained for the real case so far. We succeed in doing so by using supersymmetry techniques to express the one-point function of real Wishart correlation matrices as a twofold integral. The result can be viewed as a resummation of a series of Jack polynomials in a non-trivial case. We illustrate our formula by numerical simulations. We also rederive a known expression for the one-point function of complex Wishart correlation 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

Fig. 1
Fig. 2

Similar content being viewed by others

References

  1. Guhr, T., Müller-Groeling, A., Weidenmüller, H.A.: Phys. Rep. 299, 189 (1998)

    Article  MathSciNet  ADS  Google Scholar 

  2. Tulino, A., Verdú, S.: Found. Trends Commun. Inf. Theory 1(1), 1 (2004)

    Article  Google Scholar 

  3. Haake, F.: Quantum Signatures of Chaos, 2nd edn. Springer, Berlin (2004)

    Google Scholar 

  4. Voit, J.: The Statistical Mechanics of Financial Markets, 3rd edn. Springer, Berlin (2005)

    MATH  Google Scholar 

  5. Laurent, L., Cizeau, P., Bouchaud, J.P., Potters, M.: Phys. Rev. Lett. 83, 1467 (1999)

    Article  ADS  Google Scholar 

  6. Sprik, R., Tourin, A., de Rosny, J., Fink, M.: Phys. Rev. B 78, 012202 (2008)

    Article  ADS  Google Scholar 

  7. Muirhead, R.J.: Aspects of Multivariate Statistical Theory, 1st edn. Wiley, New York (1982)

    Book  MATH  Google Scholar 

  8. Alfano, G., Tulino, A., Lozano, A., Verdú, S.: In: Proc. IEEE 8 Int. Symp. on Spread Spectrum Tech. and Applications (ISSSTA’04), p. 515. IEEE, Bellingham (2004)

    Google Scholar 

  9. Silverstein, J.W.: J. Multivar. Anal. 55, 331 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Vinayak, Pandey, A.: Phys. Rev. E 81(3), 036202 (2010)

    Article  ADS  Google Scholar 

  11. Efetov, K.B.: Supersymmetry in Disorder and Chaos, 1st edn. Cambridge University Press, Cambridge (1997)

    MATH  Google Scholar 

  12. Verbaarschot, J.J.M.: AIP Conf. Proc. 744, 277 (2004)

    Article  MathSciNet  ADS  Google Scholar 

  13. Recher, C., Kieburg, M., Guhr, T.: Phys. Rev. Lett. 105(24), 244101 (2010)

    Article  ADS  Google Scholar 

  14. Guhr, T.: J. Phys. A 39, 13191 (2006)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Kieburg, M., Grönqvist, J., Guhr, T.: J. Phys. A 42, 275205 (2009)

    Article  MathSciNet  ADS  Google Scholar 

  16. Sommers, H.-J.: Acta Phys. Pol., B 38, 1001 (2007)

    MathSciNet  Google Scholar 

  17. Littelmann, P., Sommers, H.-J., Zirnbauer, M.R.: Commun. Math. Phys. 283, 343 (2008)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  18. Kieburg, M., Sommers, H.J., Guhr, T.: J. Phys. A 42, 275206 (2009)

    Article  MathSciNet  Google Scholar 

  19. Verbaarschot, J.J.M.: Phys. Rev. Lett. 72, 2531 (1994)

    Article  MathSciNet  ADS  Google Scholar 

  20. Harish-Chandra: Am. J. Math. 80, 241 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  21. Itzykson, C., Zuber, J.B.: J. Math. Phys. 21, 411 (1980)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  22. Efetov, K.B., Schwiete, G., Takahashi, K.: Phys. Rev. Lett. 92, 026807 (2004)

    Article  ADS  Google Scholar 

  23. Verbaarschot, J.J.M., Weidenmüller, H.A., Zirnbauer, M.R.: Phys. Rep. 129, 367 (1985)

    Article  MathSciNet  ADS  Google Scholar 

  24. Guhr, T.: In: Akemann, G., Baik, J., Di Francesco, P. (eds.) The Oxford Handbook of Random Matrix Theory. Chapter 7: Supersymmetry, pp. 135–154. Oxford University Press, Oxford (2011). arXiv:1005.0979

    Google Scholar 

  25. Berezin, F.A.: Introduction to Superanalysis. Reidel, Dordrecht (1987)

    MATH  Google Scholar 

  26. Zirnbauer, M.R.: J. Math. Phys. 37, 4986 (1996)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  27. Fyodorov, Y.V.: Nucl. Phys. B 621, 643 (2002)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  28. Wolfram Research Inc.: Mathematica version 7.0 (2008)

  29. Weyl, H.: The Classical Groups: Their Invariants and Representations. Princeton University Press, Princeton (1939)

    Google Scholar 

Download references

Acknowledgements

We thank R. Sprik for fruitful discussions on the theoretical and experimental aspects of the one-point function. We are also grateful to A. Hucht, H. Kohler and R. Schäfer for helpful comments. One of us (TG) thanks the organizers and participants of the Program in 2006 on “High Dimensional Inference and Random Matrices” at SAMSI, Research Triangle Park, North Carolina (USA), for drawing his attention to this problem. We acknowledge support from the German Research Council (DFG) via the Sonderforschungsbereich-Transregio 12 “Symmetries and Universality in Mesoscopic Systems”.

Author information

Authors and Affiliations

  1. Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco Madrid, 28049, Madrid, Spain

    Christian Recher

  2. Department of Physics and Astronomy, State University of New York, Stony Brook, NY, 11794-3800, USA

    Mario Kieburg

  3. Fakultät für Physik, Universität Duisburg-Essen, Lotharstraße 1, 47048, Duisburg, Germany

    Thomas Guhr

  4. Institut für Theoretische Physik, Universität zu Köln, Zülpicher Straße 77, 50937, Köln, Germany

    Martin R. Zirnbauer

Authors
  1. Christian Recher
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mario Kieburg
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Thomas Guhr
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Martin R. Zirnbauer
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Christian Recher.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Recher, C., Kieburg, M., Guhr, T. et al. Supersymmetry Approach to Wishart Correlation Matrices: Exact Results. J Stat Phys 148, 981–998 (2012). https://doi.org/10.1007/s10955-012-0567-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10955-012-0567-x

Keywords

Search

Navigation