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

Abstract

We considered N × N Wishart ensembles in the class W C ( Σ N , M ) (complex Wishart matrices with M degrees of freedom and covariance matrix Σ N ) such that N 0 eigenvalues of Σ N are 1 and N 1 = N − N 0 of them are a . We studied the limit as M , N , N 0 and N 1 all go to infinity such that N M → c , N 1 N → β and 0 < c , β < 1 . In this case, the limiting eigenvalue density can either be supported on 1 or 2 disjoint intervals in R + , 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.