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

Andrew Campbell, Sean O'Rourke, David Renfrew

Published 2023-07-21Version 1

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.