Distribution of the Smallest Eigenvalue in Complex and Real Correlated Wishart Ensembles

Understand

For the correlated Gaussian Wishart ensemble we compute the distribution of the smallest eigenvalue and a related gap probability.We obtain exact results for the complex (\beta=2) and for the real case (\beta=1).

• For a particular set of empirical correlation matrices we find universality in the spectral density, for both real and complex ensembles and all kinds of rectangularity.
• We calculate the asymptotic and universal results for the gap probability and the distribution of the smallest eigenvalue.
• We use the Supersymmetry method, in particular the generalized Hubbard-Stratonovich transformation and superbosonization.