Correlation kernels for sums and products of random matrices

Abstract

Let $X$ 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 $GX$ and $TX$, where $G$ is a complex Ginibre matrix and $T$ is a truncated unitary matrix. We also consider the product of $X$ 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 $H+M$ where $H$ is a GUE matrix and $M$ is a random matrix whose eigenvalue density is a polynomial ensemble. We show that the eigenvalues of $H+M$ 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.

33 pages, some changes suggested by the referee is made and some references are added

References in corpus (6)

• Recent exact and asymptotic results for products of independent random matrices
• Airy kernel with two sets of parameters in directed percolation and random matrix theory
• Singular value statistics of matrix products with truncated unitary matrices
• Determinantal structures in the O'Connell-Yor directed random polymer model
• Muttalib--Borodin ensembles in random matrix theory --- realisations and correlation functions
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices

Cited by in corpus (3)

• Products of Random Matrices from Polynomial Ensembles
• Random matrix ensembles involving Gaussian Wigner and Wishart matrices, and biorthogonal structure
• Bulk and soft-edge universality for singular values of products of Ginibre random matrices