Abstracts > Arizmendi Echegaray Octavio

Joint work with Andrew Campbell and Katsunori Fujie

Hoskins and Steinerberger (2022) showed that repeated differentiation of a random polynomial with i.i.d. mean-zero variance-one roots, followed by suitable rescaling converges to a Hermite polynomial. In this talk, I will present joint work with Andrew Campbell and Katsunori Fujie, where we extend their result using the framework of finite free probability. First we establish central limit theorems describing the fluctuations of both the polynomials and their roots around the deterministic Hermite limits. Second, we relax the finite variance assumption on the roots, uncovering a broader phenomenon in which Hermite polynomials are replaced by random Appell sequences naturally associated to finite free probability and infinitely divisible distributions. Throughout, finite free cumulants serve as the key tool enabling concise proofs.