Critical points of random polynomials and finite free cumulants

TL;DR

This paper extends a known result about the behavior of roots of random polynomials under differentiation using finite free probability, establishing CLTs and generalizing to cases without finite second moments.

Contribution

It proves central limit theorems for fluctuations of roots and polynomials, and generalizes the result to broader distributions using finite free cumulants.

Findings

CLTs for root fluctuations around deterministic limits

Extension to distributions without finite second moments

Description of generalized polynomial sequences via finite free probability

Abstract

A result of Hoskins and Steinerberger [Int. Math. Res. Not., (13):9784-9809, 2022] states that repeatedly differentiating a random polynomials with independent and identically distributed mean zero and variance one roots will result, after an appropriate rescaling, in a Hermite polynomial. We use the theory of finite free probability to extend this result in two natural directions: (1) We prove central limit theorems for the fluctuations around these deterministic limits for the polynomials and their roots. (2) We consider a generalized version of the Hoskins and Steinerberger result by removing the finite second moment assumption from the roots. In this case the Hermite polynomials are replaced by a random Appell sequence conveniently described through finite free probability and an infinitely divisible distribution. We use finite free cumulants to provide compact proofs of our main results with little prerequisite knowledge of free probability required.