The fractional free convolution of $R$-diagonal elements and random polynomials under repeated differentiation

TL;DR

This paper extends free convolution to fractional powers for $R$-diagonal elements and links it to the roots of random polynomials under repeated differentiation, revealing CLT-like behavior and stable distributions.

Contribution

It introduces a fractional free convolution for $R$-diagonal elements and connects it to the asymptotic behavior of polynomial roots under differentiation.

Findings

Fractional free convolution generalizes existing free convolution for $R$-diagonal elements.

Roots of random polynomials exhibit CLT-like behavior under repeated differentiation.

Stable distributions emerge in the asymptotic regime as the derivative proportion approaches one.

Abstract

We extend the free convolution of Brown measures of $R$-diagonal elements introduced by K\"{o}sters and Tikhomirov [Probab. Math. Statist. 38 (2018), no. 2, 359--384] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.