Finite free convolutions of polynomials

TL;DR

This paper explores three polynomial convolutions within free probability, linking them to random matrix theory, proving their real-rootedness, and establishing bounds on their roots, with a novel focus on asymmetric convolutions.

Contribution

It introduces the asymmetric additive convolution and connects all three convolutions to unitarily invariant random matrices, expanding the theoretical framework.

Findings

Proves all three convolutions produce real-rooted polynomials.

Establishes bounds on the roots of these polynomials.

Connects convolutions with sums and products of random matrices.

Abstract

We study three convolutions of polynomials in the context of free probability theory. We prove that these convolutions can be written as the expected characteristic polynomials of sums and products of unitarily invariant random matrices. The symmetric additive and multiplicative convolutions were introduced by Walsh and Szeg\"o in different contexts, and have been studied for a century. The asymmetric additive convolution, and the connection of all of them with random matrices, is new. By developing the analogy with free probability, we prove that these convolutions produce real rooted polynomials and provide strong bounds on the locations of the roots of these polynomials.