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

Andrew Campbell, Sean O'Rourke, David Renfrew

TL;DR

This work proves Cosine Universality for a broad class of even entire functions with only real zeros by recasting repeated differentiation in the language of finite free probability. It delivers three universality principles—Cosine, Laguerre, and Hermite—via finite free limit theorems: law of large numbers, central limit theorem, and Poisson limit theorem for sequences of real-rooted polynomials under differentiation. The methods connect Jensen polynomials to finite free convolutions, and extend the Laguerre–Pólya framework to infinite-root scenarios, with applications to random matrices and special functions. The results offer a probabilistic, operator-theoretic lens on the universal spacing of zeros under differentiation and provide rigorous limit theorems under mild root-density hypotheses.

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.

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

TL;DR

This work proves Cosine Universality for a broad class of even entire functions with only real zeros by recasting repeated differentiation in the language of finite free probability. It delivers three universality principles—Cosine, Laguerre, and Hermite—via finite free limit theorems: law of large numbers, central limit theorem, and Poisson limit theorem for sequences of real-rooted polynomials under differentiation. The methods connect Jensen polynomials to finite free convolutions, and extend the Laguerre–Pólya framework to infinite-root scenarios, with applications to random matrices and special functions. The results offer a probabilistic, operator-theoretic lens on the universal spacing of zeros under differentiation and provide rigorous limit theorems under mild root-density hypotheses.

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.
Paper Structure (21 sections, 20 theorems, 144 equations, 1 figure)

This paper contains 21 sections, 20 theorems, 144 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Universal limits of differentiation for the Laguerre--Pólya class
  3. Jensen polynomials and the universality principles
  4. Main results
  5. Results on the universality principles
  6. Finite free limit theorems of differentiation
  7. A note on complex roots
  8. Novelties of our results and techniques
  9. Background on finite free probability theory
  10. Proofs of the theorems in Section \ref{['sec: main results uni principles']}
  11. Proof of Theorem \ref{['thm:Laguerre uni']}
  12. Proof of Theorem \ref{['thm:Hermite uni']}
  13. Proof of Theorem \ref{['thm:Cosine uni']}
  14. Proofs of the theorems in Section \ref{['sec:finite free limit theorems']}
  15. Law of large numbers and the proof of Theorem \ref{['thm:Law of large numbers']}
  16. ...and 6 more sections

Key Result

Proposition 1.5

Let $\{J_{d,0}\}_{d=0}^{\infty}$ be the even Jensen polynomials of $\cos(z)$ of shift $0$ (so that $J_{d,0}$ is a polynomial of degree $d$). Then and where $\mathop{\mathrm{He}}\nolimits_{2d}$ and $L_{d}^{\left(-\frac{1}{2}\right)}$ are defined in eq:Hermite def and eq:Laguerre -1/2 def, respectively.

Figures (1)

  • Figure 1: Derivatives of $p_{130}$ (blue) compared to the Hermite polynomials (orange).

Theorems & Definitions (36)

  • Definition 1.1
  • Definition 1.2
  • Proposition 1.5
  • Lemma 2.2
  • Theorem 2.3: Cosine Universality for even functions
  • Theorem 2.4: Laguerre Universality
  • Theorem 2.5: Hermite Universality
  • Remark 2.6
  • Theorem 2.7: Law of large numbers
  • Theorem 2.8: Central limit theorem
  • ...and 26 more