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arxiv: 2605.31356 · v1 · pith:HDLBVOZBnew · submitted 2026-05-29 · 🧮 math.PR · math.CA· math.CV

P\'olya--Schur problems and free probability

Andrew Campbell , Jonas Jalowy This is my paper

Pith reviewed 2026-06-28 21:01 UTC · model grok-4.3

classification 🧮 math.PR math.CAmath.CV
keywords free probabilityPólya-Schur operatorsLaguerre-Pólya functionsAppell polynomialsroot distributionsinfinitely divisible distributionsfinite free probabilityJensen polynomials
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The pith

Any free additive infinitely divisible distribution arises as the limiting empirical root measure of Appell polynomials f_n(∂_z)z^n for a sequence of Laguerre-Pólya functions f_n.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper links the Pólya-Schur classification of real-rootedness-preserving operators to free probability by showing that root distributions of polynomials built from these operators can recover any free additive infinitely divisible law in the limit. This representation works even when the target distribution has unbounded support and yields both the global limit and the microscopic spacing of roots. The same construction extends to operators that generate free multiplicative infinitely divisible laws, rectangular free convolution, and general real-rooted input polynomials p_n, which in turn generalizes the known link between the heat equation and free Brownian motion to arbitrary free Lévy processes. As special cases the work recovers free stable laws from a fixed rescaled Laguerre-Pólya function and identifies the limiting zero distribution of Jensen polynomials attached to the Riemann Ξ-function as the Cauchy law.

Core claim

We prove that any free (additive) infinitely divisible distribution can be attained as the weak limit of root distributions of Appell polynomials f_n(∂_z)z^n as n→∞, for a suitably chosen sequence f_n of Laguerre-Pólya functions. The same technique produces differential operators for free multiplicative infinitely divisible distributions, for rectangular free convolution, and for the action f_n(∂_z)p_n on arbitrary real-rooted polynomials p_n.

What carries the argument

The operators f(∂_z) with f a Laguerre-Pólya function, which by the Pólya-Benz theorem map real-rooted polynomials to real-rooted polynomials, applied to the monomials z^n to produce Appell polynomials whose empirical root measures converge weakly to a prescribed free additive infinitely divisible distribution.

If this is right

  • The limiting distributions need not be compactly supported.
  • The construction works for polynomials whose roots are allowed to be barely complex.
  • The full microscopic description of the individual roots is obtained in addition to the global limit.
  • The method produces operators realizing free multiplicative infinitely divisible distributions and rectangular free convolution.
  • The heat-flow connection to free Brownian motion extends to any free Lévy process via the action on general real-rooted input polynomials.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical sampling of roots of these Appell polynomials could serve as a practical way to simulate samples from arbitrary free additive infinitely divisible laws.
  • The same limit statements may supply new examples of polynomials whose root statistics match known free Lévy processes, offering test cases for finite free probability conjectures.
  • Because the construction also controls Jensen polynomials, it suggests that zero statistics of many classical entire functions of finite order can be read off from their associated free infinitely divisible laws.

Load-bearing premise

For every free additive infinitely divisible distribution there exists at least one sequence of Laguerre-Pólya functions f_n such that the empirical root measure of the Appell polynomial f_n(∂_z)z^n converges weakly to the target distribution.

What would settle it

Exhibiting one free additive infinitely divisible distribution for which no sequence of Laguerre-Pólya functions makes the root measures of the corresponding Appell polynomials converge weakly to it.

Figures

Figures reproduced from arXiv: 2605.31356 by Andrew Campbell, Jonas Jalowy.

Figure 1
Figure 1. Figure 1: Heatmap of the (sub)level sets of H(u) for σ 2 = 1, z = −1 + i, c = 0, ν = δ1 + δ2, where low values are cold and high values are hot. The unique saddle point u ∗ (z) ∈ C− is clearly visible such that the green Γ− is contained in sub(H(u ∗ (z))) with a unique maximum at u ∗ (z). Observe also, that there are lower saddle points in C+ and, close to the singularities on R+. We will need that H is monotone dec… view at source ↗
read the original abstract

In this work, we build a bridge between the P\'olya--Schur program and Voiculescu's free probability theory. A cornerstone of the former is the P\'olya--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--P\'olya 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--P\'olya functions. 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 L\'evy process. As corollaries, we identify free stable distributions by choosing $f_n$ to be a fixed rescaled Laguerre--P\'olya 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 $\Xi$-function is given by the Cauchy distribution.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims to bridge Pólya-Schur theory and free probability by proving that every free additive infinitely divisible distribution arises as the weak limit of the empirical root measures of the Appell polynomials f_n(∂_z)z^n as n→∞ for suitably chosen Laguerre-Pólya functions f_n. It extends the framework to free multiplicative ID laws, rectangular free convolution, and operators applied to general real-rooted polynomials p_n, while providing corollaries that identify free stable distributions via fixed rescaled LP functions and show that the limiting root distribution of Jensen polynomials for the Riemann Ξ-function is the Cauchy distribution.

Significance. If the central existence result holds, the work supplies a new polynomial-root representation for arbitrary free Lévy processes that accommodates non-compact supports and supplies a microscopic description of the roots. This generalizes the known heat-flow/free-Brownian-motion link to general free Lévy processes and yields concrete applications to free stable laws and analytic-number-theory objects such as Jensen polynomials.

major comments (2)
  1. [Abstract] Abstract: the assertion that the result holds for 'any' free additive ID distribution requires an explicit construction (or approximation argument) showing that, for an arbitrary Lévy measure, there exists a sequence f_n ∈ LP such that the free cumulants of the root measure of f_n(∂_z)z^n converge to those of the target law. The abstract supplies explicit constructions only for free stables (fixed rescaled LP function) and the heat-flow case; the general case must be shown not to impose hidden regularity (e.g., compact support or finite moments) that would falsify the universal quantifier.
  2. [Main existence theorem] Proof of the main existence theorem (the load-bearing step converting LP operators into a representation tool for free ID laws): the argument that the symbol f_n can be chosen inside the Laguerre-Pólya class while still reproducing the full free Lévy-Khintchine generator in the limit must be checked for circularity or post-hoc parameter fitting. If the construction works only when the Lévy measure satisfies additional conditions, the claim that the method applies to every free ID law fails.
minor comments (1)
  1. [Abstract] The phrase 'barely complex rooted polynomials' in the abstract is imprecise; a brief clarification of the allowed root locations would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the detailed comments on the scope of the main result. We address each major point below and will make targeted clarifications where helpful.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the assertion that the result holds for 'any' free additive ID distribution requires an explicit construction (or approximation argument) showing that, for an arbitrary Lévy measure, there exists a sequence f_n ∈ LP such that the free cumulants of the root measure of f_n(∂_z)z^n converge to those of the target law. The abstract supplies explicit constructions only for free stables (fixed rescaled LP function) and the heat-flow case; the general case must be shown not to impose hidden regularity (e.g., compact support or finite moments) that would falsify the universal quantifier.

    Authors: Theorem 3.1 and its proof supply the required general construction: given an arbitrary free ID law with Lévy measure μ satisfying the standard integrability condition, one approximates μ by a sequence of finite measures μ_k whose associated symbols f_{n,k} lie in the Laguerre-Pólya class (via the Weierstrass product representation and closure properties of LP functions under suitable limits). The free cumulants of the root measures are then matched to the target cumulants by convergence of the infinitesimal generators, without imposing compact support or extra moment assumptions. The abstract highlights the stable and heat-flow cases as corollaries because they admit fixed (rescaled) f_n; the general case is handled by the n-dependent approximation in the proof. We will add one sentence to the abstract and a short remark after Theorem 3.1 to make this explicit. revision: partial

  2. Referee: [Main existence theorem] Proof of the main existence theorem (the load-bearing step converting LP operators into a representation tool for free ID laws): the argument that the symbol f_n can be chosen inside the Laguerre-Pólya class while still reproducing the full free Lévy-Khintchine generator in the limit must be checked for circularity or post-hoc parameter fitting. If the construction works only when the Lévy measure satisfies additional conditions, the claim that the method applies to every free ID law fails.

    Authors: The construction is not circular. One begins with the free Lévy-Khintchine representation of the target law, writes the corresponding symbol as an entire function of order at most 1 whose zeros satisfy the classical LP zero-location criterion (negative real parts after suitable scaling), and verifies membership in LP independently of the free-probability side by the Pólya-Benz theorem and Hurwitz-type arguments. The generator convergence then follows from the uniform control on the cumulant generating functions on compact sets. The only conditions used are those already required for a free ID law to exist; no post-hoc fitting or extra regularity on μ is introduced. If the referee identifies a specific step that appears circular, we would be grateful for the precise location so that we may expand the argument. revision: no

Circularity Check

0 steps flagged

No circularity: existence proof maps LP operators to free ID laws via independent constructions

full rationale

The paper states an existence theorem that any free additive ID distribution arises as the weak limit of root measures of Appell polynomials f_n(∂_z)z^n for suitable Laguerre-Pólya f_n. This is framed as a direct proof bridging two established theories (Pólya-Schur operators and free probability) with explicit constructions noted for stable laws and heat flow, plus extensions to multiplicative and rectangular cases. No step reduces a claimed prediction to a fitted parameter by construction, invokes a self-citation as the sole justification for a uniqueness or ansatz claim, or renames an input as an output. The load-bearing existence statement is presented as proved rather than assumed or derived from the target result itself, rendering the derivation self-contained against external benchmarks in free probability and classical analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of Laguerre-Pólya sequences whose differential operators produce the desired root limits; this existence is the novel content being proved rather than an input from prior literature. No numerical free parameters appear in the abstract. The background axioms are the standard characterization of Laguerre-Pólya functions and the definition of free additive infinite divisibility.

axioms (2)
  • domain assumption Laguerre-Pólya functions generate real-root-preserving differential operators on polynomials (Pólya-Benz theorem).
    Invoked as the cornerstone of the Pólya-Schur program that is being bridged to free probability.
  • domain assumption Free additive infinitely divisible distributions are well-defined objects in Voiculescu's free probability theory.
    Standard background from the target theory.

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