A theory of singular values for finite free probability

TL;DR

This paper develops a finite free probability theory for rectangular matrices, enabling the analysis of singular values of polynomials and establishing convergence to classical free probability results.

Contribution

It introduces a finite version of free probability for rectangular matrices, extending classical concepts to a polynomial framework with asymptotic convergence.

Findings

Replicates free probability transforms for rectangular matrices.

Establishes asymptotic convergence from finite to classical free probability.

Derives explicit classical distribution results within the new framework.

Abstract

We introduce a finite version of free probability for rectangular matrices that amounts to operations on singular values of polynomials. We show that we can replicate the transforms from free probability, and that asymptotically there is convergence from rectangular finite free probability to rectangular free probability. Lastly, we show that classical distribution results such as a law of large numbers or a central limit theorem can be made explicit in this new framework where random variables are replaced by polynomials.