Journal article
Supersymmetry Approach to Wishart Correlation Matrices: Exact Results
C Recher, M Kieburg, T Guhr, MR Zirnbauer
Journal of Statistical Physics | SPRINGER | Published : 2012
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. © 2012 Springer Science+Business Media, LLC.
Grants
Funding 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. Schafer 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".