Limiting Distributions of Spectral Radii for Product of Matrices from the Spherical Ensemble

TL;DR

This paper studies the limiting behavior of the spectral radii of products of independent matrices from the spherical ensemble, revealing convergence to Gamma-based distributions for fixed m and normal limits when m grows with n.

Contribution

It extends previous results by characterizing the spectral radius distributions for products of spherical ensemble matrices in both fixed and growing m scenarios.

Findings

Spectral radii converge to distributions of functions of Gamma variables for fixed m.

Logarithmic spectral radii tend to a normal distribution as m increases with n.

Provides a unified understanding of spectral radius behavior in matrix products from the spherical ensemble.

Abstract

Consider the product of $m$ independent $n\times n$ random matrices from the spherical ensemble for $m\ge 1$. The spectral radius is defined as the maximum absolute value of the $n$ eigenvalues of the product matrix. When $m=1$, the limiting distribution for the spectral radii has been obtained by Jiang and Qi (2017). In this paper, we investigate the limiting distributions for the spectral radii in general. When $m$ is a fixed integer, we show that the spectral radii converge weakly to distributions of functions of independent Gamma random variables. When $m=m_n$ tends to infinity as $n$ goes to infinity, we show that the logarithmic spectral radii have a normal limit.