Universality in complex Wishart ensembles for general covariance matrices with 2 distinct eigenvalues

We considered NxN Wishart ensembles in the class (complex Wishart matrices with M degrees of freedom and covariance matrix [Sigma]N) such that N0 eigenvalues of [Sigma]N are 1 and N1=N-N0 of them are a. We studied the limit as M, N, N0 and N1 all go to infinity such that , and 0<c,[beta]<1. In this case, the limiting eigenvalue density can either be supported on 1 or 2 disjoint intervals in , and a phase transition occurs when the support changes from 1 interval to 2 intervals. By using the Riemann-Hilbert analysis, we have shown that when the phase transition occurs, the eigenvalue distribution is described by the Pearcey kernel near the critical point where the support splits.