Determinantal point processes in the plane from products of random matrices

TL;DR

This paper investigates the eigenvalue distributions of various classes of random matrix products, deriving explicit formulas for their spectral densities and limiting distributions, revealing determinantal point process structures.

Contribution

It provides new explicit formulas for eigenvalue densities of products of Gaussian, rectangular, and truncated unitary matrices, expanding understanding of their spectral properties.

Findings

Eigenvalue densities are determinantal for the studied ensembles.

Explicit formulas for spectral densities of matrix products are derived.

Limiting spectral distributions are characterized for each ensemble.

Abstract

We show the density of eigenvalues for three classes of random matrix ensembles is determinantal. First we derive the density of eigenvalues of product of $k$ independent $n\times n$ matrices with i.i.d. complex Gaussian entries with a few of matrices being inverted. In second example we calculate the same for (compatible) product of rectangular matrices with i.i.d. Gaussian entries and in last example we calculate for product of independent truncated unitary random matrices. We derive exact expressions for limiting expected empirical spectral distributions of above mentioned ensembles.