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The fractional free convolution of $R$-diagonal elements and random polynomials under repeated differentiation

Andrew Campbell, Sean O'Rourke, David Renfrew

Abstract

We extend the free convolution of Brown measures of $R$-diagonal elements introduced by Kösters and Tikhomirov [Probab. Math. Statist. 38 (2018), no. 2, 359--384] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.

The fractional free convolution of $R$-diagonal elements and random polynomials under repeated differentiation

Abstract

We extend the free convolution of Brown measures of -diagonal elements introduced by Kösters and Tikhomirov [Probab. Math. Statist. 38 (2018), no. 2, 359--384] to fractional powers. We then show how this fractional free convolution arises naturally when studying the roots of random polynomials with independent coefficients under repeated differentiation. When the proportion of derivatives to the degree approaches one, we establish central limit theorem-type behavior and discuss stable distributions.
Paper Structure (28 sections, 23 theorems, 140 equations, 2 figures)

This paper contains 28 sections, 23 theorems, 140 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Free probability theory background
  3. Free probability theory background and notation
  4. The fractional free convolution for self-adjoint operators
  5. The Brown measure
  6. $R$-diagonal elements
  7. Random polynomial theory background
  8. Random polynomials with independent coefficients
  9. PDEs describing the behavior of roots under repeated differentiation
  10. Connections to free probability theory
  11. The fractional free convolution for $R$-diagonal elements
  12. Fractional free convolution powers of the Brown measure
  13. Connection between Brown measures and derivatives of random polynomials
  14. Consequences of the theory and examples
  15. Examples
  16. ...and 13 more sections

Key Result

Theorem 1.1

Let $\mu$ be a compactly supported probability measure on the real line, and let $P_n$ be the random polynomial where $X_1, X_2, \ldots$ are independent and identically distributed (iid) random variables with distribution $\mu$. For any fixed $t \in (0,1)$, the empirical root distribution of the $\lceil tn \rceil$-th derivative of $P_n((1-t)x)$ converges weakly almost surely to $\mu^{\boxplus 1/(

Figures (2)

  • Figure 1: Numerical simulations illustrating Theorem \ref{['thm:A:Kacs free convolution']} and Proposition \ref{['prop:sumsunitary']}. The figure on the left shows the radial cumulative distribution function for the eigenvalues of the sum of two independent, Haar distributed unitary random matrices. The figure on the right is constructed using the random polynomial $P_n$ given in \ref{['eq:kacmodel']} when $n = 1000$. The figure depicts the radial cumulative distribution function of the empirical root measure of the $n/2$-th derivative of $P_n$, after applying the push-forward map $\mathop{\mathrm{Sq}}\nolimits^{-1}$.
  • Figure 2: This diagram represents the relationship between the free compression of $R$-diagonal elements and repeated differentiation of random polynomials. The map $\mathop{\mathrm{Sq}}\nolimits$ on radially symmetric measures is defined before Theorem \ref{['thm:A:Kacs free convolution']}. Note, when comparing repeated differentiation directly to free compressions there is no need to include any rescaling of the roots, unlike when comparing to the convolution $\oplus$.

Theorems & Definitions (48)

  • Theorem 1.1: Hoskins--Kabluchko, Steinerberger, Arizmendi--Garza-Vargas--Perales
  • Theorem 1.2
  • Proposition 1.3: Basak--Dembo
  • Proposition 1.4: Życzkowski--Sommers, Dénes--Réffy
  • Remark 2.1
  • Remark 2.2
  • Remark 2.3
  • Remark 2.4
  • Theorem 2.5: Haagerup--Larsen, Zhong
  • Remark 2.6
  • ...and 38 more