New spectral relations between products and powers of isotropic random matrices

TL;DR

This paper establishes a spectral relation between the eigenvalue densities of products and powers of isotropic random matrices, simplifying the analysis of their spectral properties in large dimensions.

Contribution

It demonstrates that the eigenvalue density of a product of isotropic matrices equals that of a single matrix's power, extending to orthogonal ensembles and specific matrix examples.

Findings

Eigenvalue density of product equals that of power in the limit

Derived densities for products of Girko-Ginibre and truncated unitary matrices

Evidence suggests the relation holds for isotropic orthogonal ensembles

Abstract

We show that the limiting eigenvalue density of the product of n identically distributed random matrices from an isotropic unitary ensemble (IUE) is equal to the eigenvalue density of n-th power of a single matrix from this ensemble, in the limit when the size of the matrix tends to infinity. Using this observation one can derive the limiting density of the product of n independent identically distributed non-hermitian matrices with unitary invariant measures. In this paper we discuss two examples: the product of n Girko-Ginibre matrices and the product of n truncated unitary matrices. We also provide an evidence that the result holds also for isotropic orthogonal ensembles (IOE).