Large Complex Correlated Wishart Matrices: Fluctuations and Asymptotic Independence at the Edges

We study the asymptotic behavior of eigenvalues of large complex correlated Wishart matrices at the edges of the limiting spectrum. In this setting, the support of the limiting eigenvalue distribution may have several connected components. Under mild conditions for the population matrices, we show that for every generic positive edge of that support, there exists an extremal eigenvalue which converges almost surely towards that edge and fluctuates according to the Tracy-Widom law at the scale N. Moreover, given several generic positive edges, we establish that the associated extremal eigenvalue fluctuations are asymptotically independent. Finally, when the leftmost edge is the origin, we prove that the smallest eigenvalue fluctuates according to the hard-edge Tracy-Widom law at the scale N. As an application, an asymptotic study of the condition number of large correlated Wishart matrices is provided. AMS 2000 subject classification: Primary 15A52, Secondary 15A18, 60F15.