Product of Independent Cauchy-Lorentz Random Matrices

TL;DR

This paper analyzes the eigenvalue distribution of the product of multiple independent Cauchy-Lorentz random matrices, revealing a determinantal point process with a Meijer G-function weight that generalizes the single-matrix case.

Contribution

It extends the understanding of eigenvalue distributions to products of multiple Cauchy-Lorentz matrices, deriving explicit joint distributions.

Findings

Eigenvalues form a determinantal point process

Joint distribution involves a Meijer G-function

Generalizes single-matrix eigenvalue results

Abstract

We investigate the product of $n$ complex non-Hermitian, independent random matrices, each of size $N_i\times N_{i+1}$ $(i=1,...,n)$, with independent identically distributed Cauchy entries (Cauchy-Lorentz matrices). The joint probability distribution of the complex eigenvalues of the product matrix is found to be given by a determinantal point process as in the case of a single Cauchy-Lorentz matrix, but with weight given by a Meijer G-function depending on $n$ and $N_i$.