Universality for roots of derivatives of entire functions via finite free probability

TL;DR

This paper proves a universality conjecture about the roots of derivatives of entire functions, using finite free probability theory, and extends results to Jensen polynomials and Hermite universality.

Contribution

It establishes the Cosine Universality conjecture for a class of entire functions and develops finite free probability analogs of classical limit theorems.

Findings

Roots of derivatives become evenly spaced in the limit for certain entire functions.

Finite free probability analogs of LLN, CLT, and Poisson theorems are proven.

Universality results extend to Jensen polynomials and Hermite universality.

Abstract

A universality conjecture of Farmer and Rhoades [Trans. Amer. Math. Soc., 357(9):3789--3811, 2005] and Farmer [Adv. Math., 411:Paper No. 108781, 14, 2022] asserts that, under some natural conditions, the roots of an entire function should become perfectly spaced in the limit of repeated differentiation. This conjecture is known as Cosine Universality. We establish this conjecture for a class of even entire functions with only real roots which are real on the real line. Along the way, we establish a number of additional universality results for Jensen polynomials of entire functions, including the Hermite Universality conjecture of Farmer [Adv. Math., 411:Paper No. 108781, 14, 2022]. Our proofs are based on finite free probability theory. We establish finite free probability analogs of the law of large numbers, central limit theorem, and Poisson limit theorem for sequences of deterministic polynomials under repeated differentiation, under optimal moment conditions, which are of independent interest.