A non-Hermitian generalisation of the Marchenko-Pastur distribution: from the circular law to multi-criticality
We consider the complex eigenvalues of a Wishart type random matrix model X = X1X2*, where two rectangular complex Ginibre matrices X-1,X-2 of size N x (N + nu) are correlated through a non-Hermiticity parameter t is an element of [0, 1]. For general nu = O(N) and tau, we obtain the global limiting density and its support, given by a shifted ellipse. It provides a non-Hermitian generalisation of the Marchenko-Pastur distribution, which is recovered at maximal correlation X-1 = X-2 when tau = 1. The square root of the complex Wishart eigenvalues, corresponding to the nonzero complex eigenvalues of the Dirac matrix D = ((0) (X2) (X1)(0)), are supported in a domain parametrised by a quartic equation. It displays a lemniscate type transition at a critical value tc, where the interior of the spectrum splits into two connected components. At multi-criticality, we obtain the limiting local kernel given by the edge kernel of the Ginibre ensemble in squared variables. For the global statistics, we apply Frostman's equilibrium problem to the 2D Coulomb gas, whereas the local statistics follows from a saddle point analysis of the kernel of orthogonal Laguerre polynomials in the complex plane.
Springer