Product of Independent Cauchy-Lorentz Random Matrices

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 $N_i$.