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Finite Free Information Inequalities

Jorge Garza-Vargas, Nikhil Srivastava, Zachary Stier

Abstract

We develop finite free information theory for real-rooted polynomials, establishing finite free analogues of entropy and Fisher information monotonicity, as well as the Stam and entropy power inequalities. These results resolve conjectures by Shlyakhtenko and Gribinski and recover inequalities in free probability in the large-degree limit. Equivalently, our results may be interpreted as potential-theoretic inequalities for the zeros of real-rooted polynomials under differential operators which preserve real-rootedness. Our proofs leverage a new connection between score vectors and Jacobians of root maps, combined with convexity results for hyperbolic polynomials.

Finite Free Information Inequalities

Abstract

We develop finite free information theory for real-rooted polynomials, establishing finite free analogues of entropy and Fisher information monotonicity, as well as the Stam and entropy power inequalities. These results resolve conjectures by Shlyakhtenko and Gribinski and recover inequalities in free probability in the large-degree limit. Equivalently, our results may be interpreted as potential-theoretic inequalities for the zeros of real-rooted polynomials under differential operators which preserve real-rootedness. Our proofs leverage a new connection between score vectors and Jacobians of root maps, combined with convexity results for hyperbolic polynomials.
Paper Structure (27 sections, 25 theorems, 121 equations)

This paper contains 27 sections, 25 theorems, 121 equations.

Table of Contents

  1. Introduction
  2. Statement of results
  3. Related work
  4. Relation to free probability theory
  5. Free entropy.
  6. Free Fisher information.
  7. Acknowledgments.
  8. Preliminaries
  9. Roots of derivatives
  10. The heat flow
  11. Hyperbolic polynomials
  12. The finite free convolution
  13. Relating score vectors
  14. Blachman's proof
  15. Jacobians of root maps
  16. ...and 12 more sections

Key Result

Theorem 1.3

Let $p(x)$ be a real-rooted polynomial of degree $n$. If $\widetilde{p'}(x)$ is the rescaling of $p'(x)$ satisfying $\mathrm{Var}\left(\widetilde{p'}\right)=\mathrm{Var}(p)$, then

Theorems & Definitions (62)

  • Definition 1.1: Score, Fisher information, and entropy
  • Remark 1.2: Discriminant and Coulomb potential
  • Theorem 1.3: Finite free Fisher information monotonicity
  • Theorem 1.4: Finite free Stam inequality
  • Theorem 1.5: Mononicity of finite free entropy under differentiation
  • Theorem 1.6: Finite free entropy power inequality
  • Remark 1.7: Brunn--Minkowski inequality
  • Definition 2.1: Vectors of roots
  • Definition 2.2: Map for roots of the derivative
  • proof
  • ...and 52 more