Lyapunov exponents for products of rectangular real, complex and quaternionic Ginibre matrices

TL;DR

This paper analyzes the eigenvalue distributions of products of rectangular Ginibre matrices across real, complex, and quaternionic classes, revealing universal behaviors and phase distributions in the large matrix product limit.

Contribution

It provides a detailed characterization of eigenvalue joint densities and phase distributions for products of Ginibre matrices, extending understanding across different algebraic classes.

Findings

Eigenvalues form a permanental point process in the large product limit

Eigenvalue moduli become uncorrelated and log-normal

Phase distributions depend on matrix class (real, complex, quaternionic)

Abstract

We study the joint density of eigenvalues for products of independent rectangular real, complex and quaternionic Ginibre matrices. In the limit where the number of matrices tends to infinity, it is shown that the joint probability density function for the eigenvalues forms a permanental point process for all three classes. The moduli of the eigenvalues become uncorrelated and log-normal distributed, while the distribution for the phases of the eigenvalues depends on whether real, complex or quaternionic Ginibre matrices are considered. In the derivation for a product of real matrices, we explicitly use the fact that all eigenvalues become real when the number of matrices tends to infinity. Finally, we compare our results with known results for the Lyapunov exponents as well as numerical simulations.