Regularity and Convergence Properties of Finite Free Convolutions

TL;DR

This paper explores the regularity and convergence properties of finite free convolutions, establishing inequalities and conditions that lead to their weak and Kolmogorov distance convergence to infinite free convolutions as polynomial degree increases.

Contribution

It introduces new triangle inequalities and atom conditions for finite free convolutions, and proves their convergence to infinite free convolutions without compactness assumptions.

Findings

Established triangle inequalities for finite free convolutions.

Derived necessary and sufficient conditions for atoms of probability measures.

Proved weak convergence and convergence in Kolmogorov distance of finite to infinite free convolutions.

Abstract

Finite free convolutions, $\boxplus_d$ and $\boxtimes_d$, are binary operations on polynomials of degree $d$ that are central to finite free probability, a developing field at the intersection of free probability and the geometry of polynomials. Motivated by established regularities in free probability, this paper investigates analogous regularities for finite free convolutions. Key findings include triangle inequalities for these convolutions and necessary and sufficient conditions regarding atoms of probability measures. Applications of these results include proving the weak convergence of $\boxplus_d$ and $\boxtimes_d$ to their infinite counterparts $\boxplus$ and $\boxtimes$ as $d \to \infty$, without compactness assumptions. Furthermore, this weak convergence is strengthened to convergence in Kolmogorov distance.