Correlation kernels for sums and products of random matrices

Abstract

Let [Formula: see text] be a random matrix whose squared singular value density is a polynomial ensemble. We derive double contour integral formulas for the correlation kernels of the squared singular values of [Formula: see text] and [Formula: see text], where [Formula: see text] is a complex Ginibre matrix and [Formula: see text] is a truncated unitary matrix. We also consider the product of [Formula: see text] and several complex Ginibre/truncated unitary matrices. As an application, we derive the precise condition for the squared singular values of the product of several truncated unitary matrices to follow a polynomial ensemble. We also consider the sum [Formula: see text] where [Formula: see text] is a GUE matrix and [Formula: see text] is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of [Formula: see text] follow a polynomial ensemble whose correlation kernel can be expressed as a double contour integral. As an application, we point out a connection to the two-matrix model.