# Empirical Distribution of Scaled Eigenvalues for Product of Matrices from the Spherical Ensemble

## TL;DR

This paper studies the limiting distribution of scaled eigenvalues for products of matrices from the spherical ensemble, especially when the number of matrices varies with the size of the matrices.

## Contribution

It extends previous results by analyzing the empirical spectral distribution when the number of matrices grows with the matrix size.

## Key findings

- Derived the limiting empirical spectral distribution for varying m_n
- Generalized previous fixed-m results to m_n sequences
- Provided insights into eigenvalue behavior for large matrix products

## Abstract

Consider the product of $m$ independent $n\times n$ random matrices from the spherical ensemble for $m\ge 1$. The empirical distribution based on the $n$ eigenvalues of the product is called the empirical spectral distribution. Two recent papers by G\"otze, K\"osters and Tikhomirov (2015) and Zeng (2016) obtain the limit of the empirical spectral distribution for the product when $m$ is a fixed integer. In this paper, we investigate the limiting empirical distribution of scaled eigenvalues for the product of $m$ independent matrices from the spherical ensemble in the case when $m$ changes with $n$, that is, $m=m_n$ is an arbitrary sequence of positive integers.