Asymptotic root distribution of polynomials under repeated polar differentiation

TL;DR

This paper investigates the asymptotic distribution of roots of polynomials under repeated polar differentiation, revealing connections to free convolution, M"obius transforms, and introducing a new family of measure operations with applications to classical distributions.

Contribution

It introduces a novel family of measure operations related to polar derivatives, establishing their algebraic properties and connections to free probability.

Findings

The limiting root distribution is described via fractional free convolution.

The measure operations form a semigroup with specific algebraic properties.

Examples include distributions like Marchenko-Pastur and Cauchy that behave well under these operations.

Abstract

Given a sequence of real rooted polynomials $\{p_n\}_{n\geq 1}$ with a fixed asymptotic root distribution, we study the asymptotic root distribution of the repeated polar derivatives of this sequence. This limiting distribution can be seen as the result of fractional free convolution and pushforward maps along M\"obius transforms for distributions. This new family of operations on measures forms a semigroup and satisfy some other nice properties. Using the fact that polar derivatives commute with one another, we obtain a non-trivial commutation relation between these new operations. We also study a notion of polar free infinite divisibility and construct Belinschi-Nica type semigroups. Finally, we provide some interesting examples of distributions that behave nicely with respect to these new operations, including the Marchenko-Pastur and the Cauchy distributions.