Spectral correlation functions of the sum of two independent complex Wishart matrices with unequal covariances (Q2816809)

The authors compute the joint probability distribution of the eigenvalues of the sum of two independent complex Wishart matrices in the half degenerate case, i.e., when the correlation matrix appearing in the definition of the probability distribution of the first summand is proportional to the identity and the correlation matrix appearing in the definition of the probability distribution of the second summand is diagonal. Indeed, in this case, the joint probability distribution of the eigenvalues is described by a determinantal point process defined by a kernel given in terms of bi-orthogonal functions. The kernel is then computed making use of integration over Grassmann variables as well as other computing tools commonly used in theoretical physics. The same technique allows also to compute the generating function for the \(k\)-point correlation function, and thus the joint probability distribution of the eigenvalues, in the general case. Asymptotic formulas for large ensembles are also given.