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

Abstract. For the correlated Gaussian Wishart ensemble we compute the dis-tribution of the smallest eigenvalue and a related gap probability.We obtain exact results for the complex (β = 2) and for the real case (β = 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