Eigenvalue Densities of Real and Complex Wishart Correlation Matrices
Christian Recher, Mario Kieburg, Thomas Guhr
PHYSICAL REVIEW LETTERS | AMER PHYSICAL SOC | Published : 2010
Wishart correlation matrices are the standard model for the statistical analysis of time series. The ensemble averaged eigenvalue density is of considerable practical and theoretical interest. For complex time series and correlation matrices, the eigenvalue density is known exactly. In the real case, a fundamental mathematical obstacle made it forbiddingly complicated to obtain exact results. We use the supersymmetry method to fully circumvent this problem. We present an exact formula for the eigenvalue density in the real case in terms of twofold integrals and finite sums.
Awarded by Deutsche Forschungsgemeinschaft
We thank R. Sprik for fruitful discussions as well as A. Hucht, H. Kohler, and R. Schafer for helpful comments. One of us (T. G.) greatly benefitted from the Program on High Dimensional Inference and Random Matrices in 2006 at SAMSI, Research Triangle Park, North Carolina (USA). We acknowledge support from Deutsche Forschungsgemeinschaft (Sonderforschungsbereich Transregio 12).