Eigenvalues of Product of Ginibre Ensembles and Their Inverses and that of Truncated Haar Unitary Matrices and Their Inverses

Abstract: Consider two types of products of independent random matrices, including products of Ginibre matrices and inverse Ginibre matrices and products of truncated Haar unitary matrices and inverse truncated Haar matrices. Each product matrix has $m$ multiplicands of $n$ by $n$ square matrices, and the empirical distribution based on the $n$ eigenvalues of the product matrix is called empirical spectral distribution of the matrix. In this paper, we investigate the limiting empirical spectral distribution of the product matrices when $n$ tends to infinity and $m$ changes with $n$. For properly scaled eigenvalues for two types of the product matrices, we obtain the necessary and sufficient conditions for the convergence of the empirical spectral distributions.