Permanental processes from products of complex and quaternionic induced Ginibre ensembles

TL;DR

This paper studies the eigenvalue distributions of products of complex and quaternionic induced Ginibre matrices, revealing they form permanental processes and deriving exact and asymptotic results for hole and overcrowding probabilities.

Contribution

It extends the understanding of eigenvalue processes to products of complex and quaternionic matrices, generalizing previous results for single matrices.

Findings

Eigenvalues form a permanental process for products of matrices.

Exact formulas for hole probabilities in eigenvalue distributions.

Asymptotic expansions for overcrowding probabilities.

Abstract

We consider products of independent random matrices taken from the induced Ginibre ensemble with complex or quaternion elements. The joint densities for the complex eigenvalues of the product matrix can be written down exactly for a product of any fixed number of matrices and any finite matrix size. We show that the squared absolute values of the eigenvalues form a permanental process, generalising the results of Kostlan and Rider for single matrices to products of complex and quaternionic matrices. Based on these findings, we can first write down exact results and asymptotic expansions for the so-called hole probabilities, that a disc centered at the origin is void of eigenvalues. Second, we compute the asymptotic expansion for the opposite problem, that a large fraction of complex eigenvalues occupies a disc of fixed radius centered at the origin; this is known as the overcrowding problem. While the expressions for finite matrix size depend on the parameters of the induced ensembles, the asymptotic results agree to leading order with previous results for products of square Ginibre matrices.