---
_id: '66642'
abstract:
- lang: eng
  text: "In this work, we build a bridge between the Pólya--Schur program and Voiculescu's
    free probability theory. A cornerstone of the former is the Pólya--Benz Theorem,
    classifying a central family of real-root preserving operators on the space of
    polynomials, as those given by $f(\\partial_z)$ for a Laguerre--Pólya function
    $f$ and the derivative operator $\\partial_{z}$. We prove that any free (additive)
    infinitely divisible distribution can be attained as the weak limit of root distributions
    of Appell polynomials $f_n(\\partial_z)z^n$ as $n\\to\\infty$, for a suitably
    chosen sequence $f_n$ of Laguerre--Pólya functions.\r\n  Such questions on the
    (global) limiting distributions of real rooted polynomials belong to the active
    research area of finite free probability. In contrast to its standard tools, our
    approach allows for non-compactly supported limiting distributions, (barely) complex
    rooted polynomials and even provides the full microscopic description of the roots.
    Moreover, we extend our results to differential operators generating free multiplicative
    infinitely divisible distributions, to the rectangular free convolution, and to
    $f_n(\\partial_z)p_n$ for real rooted polynomials $p_n$, implying a generalization
    of the recent connections between the heat flow and free Brownian motion to any
    free Lévy process.\r\n  As corollaries, we identify free stable distributions
    by choosing $f_n$ to be a fixed rescaled Laguerre--Pólya function, and we prove
    various convergence results on the zero distributions of Jensen polynomials, e.g.
    the limiting root distribution of Jensen polynomial of the Riemann $Ξ$-function
    is given by the Cauchy distribution."
author:
- first_name: Andrew
  full_name: Campbell, Andrew
  last_name: Campbell
- first_name: Jonas
  full_name: Jalowy, Jonas
  id: '113768'
  last_name: Jalowy
  orcid: 0000-0001-9624-2685
citation:
  ama: Campbell A, Jalowy J. Pólya--Schur problems and free probability. <i>arXiv:260531356</i>.
    Published online 2026.
  apa: Campbell, A., &#38; Jalowy, J. (2026). Pólya--Schur problems and free probability.
    In <i>arXiv:2605.31356</i>.
  bibtex: '@article{Campbell_Jalowy_2026, title={Pólya--Schur problems and free probability},
    journal={arXiv:2605.31356}, author={Campbell, Andrew and Jalowy, Jonas}, year={2026}
    }'
  chicago: Campbell, Andrew, and Jonas Jalowy. “Pólya--Schur Problems and Free Probability.”
    <i>ArXiv:2605.31356</i>, 2026.
  ieee: A. Campbell and J. Jalowy, “Pólya--Schur problems and free probability,” <i>arXiv:2605.31356</i>.
    2026.
  mla: Campbell, Andrew, and Jonas Jalowy. “Pólya--Schur Problems and Free Probability.”
    <i>ArXiv:2605.31356</i>, 2026.
  short: A. Campbell, J. Jalowy, ArXiv:2605.31356 (2026).
date_created: 2026-08-04T06:07:17Z
date_updated: 2026-08-04T06:07:45Z
department:
- _id: '94'
external_id:
  arxiv:
  - '2605.31356'
language:
- iso: eng
publication: arXiv:2605.31356
status: public
title: Pólya--Schur problems and free probability
type: preprint
user_id: '113768'
year: '2026'
...