Free Convolution Powers via Roots of Polynomials

TL;DR

This paper provides an elementary polynomial-based method to describe free convolution powers of probability measures, linking free probability theory with polynomial root analysis.

Contribution

It introduces a novel polynomial and root-based approach to characterize free convolution powers, simplifying previous complex analytical methods.

Findings

Polynomial roots characterize free convolution powers

Bridges free probability with polynomial asymptotics

Simplifies analysis of free convolution powers

Abstract

Let $\mu$ be a compactly supported probability measure on the real line. Bercovici-Voiculescu and Nica-Speicher proved the existence of a free convolution power $\mu^{\boxplus k}$ for any real $k \geq 1$. The purpose of this short note is to give an elementary description of $\mu^{\boxplus k}$ in terms of of polynomials and roots of their derivatives. This bridge allows us to switch back and forth between free probability and the asymptotic behavior of polynomials.