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arxiv: 2604.07647 · v1 · submitted 2026-04-08 · 🧮 math.PR

Limiting Root Distribution of Random Log-concave Polynomials

Ohad Noy Feldheim , Arnab Sen This is my paper

Pith reviewed 2026-05-10 16:51 UTC · model grok-4.3

classification 🧮 math.PR
keywords random polynomialslog-concave sequencesempirical root distributionKac polynomialsbeta modeluniform modelcomplex zeros
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The pith

Random log-concave polynomials have roots that converge to the unit circle in one model and fill the plane according to a new symmetric density in the other.

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

The paper defines two probabilistic constructions that generate random polynomials whose coefficients form log-concave sequences. In the uniform model the roots of high-degree instances converge in distribution to the uniform measure on the unit circle, recovering the same limiting law found for classical Kac polynomials. In the beta model an exponential scaling of the coefficients strengthens the log-concavity constraint and produces a rotationally symmetric limiting density that is absolutely continuous with respect to area measure on the complex plane. These results show that log-concavity alone does not dictate a single root law; the precise strength of the constraint matters for the asymptotic geometry.

Core claim

In the uniform model, the empirical root distribution converges to the uniform probability measure on the unit circle, placing the model in the same universality class as classical Kac polynomials. In the beta model log-concavity is amplified through exponential scaling of the coefficients, leading to a new limiting distribution that is rotationally symmetric and absolutely continuous with respect to Lebesgue measure on the plane.

What carries the argument

The uniform and beta probabilistic models that sample random log-concave coefficient sequences, together with the empirical measures of their complex roots.

If this is right

  • The uniform model belongs to the same universality class as Kac polynomials for zero distributions.
  • The beta model produces a limiting root law that is absolutely continuous on the plane rather than supported on the circle.
  • Strengthening the log-concavity constraint via coefficient scaling changes the support and type of the limiting root measure.
  • Both limits are rotationally symmetric, indicating that the underlying coefficient laws preserve circular symmetry in the roots.

Where Pith is reading between the lines

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

  • The results suggest that the degree of log-concavity can be tuned to move roots from the boundary into the interior of the disk.
  • Similar constructions could be applied to other classes of constrained random polynomials to test whether the same dichotomy appears.
  • The beta-model density may admit an explicit formula in polar coordinates that could be derived from the coefficient scaling.

Load-bearing premise

The two explicit constructions actually produce coefficient sequences that satisfy the log-concavity inequality and that the root asymptotics are controlled solely by these coefficient distributions.

What would settle it

Numerical computation of the empirical root measure for degree-1000 polynomials in the uniform model that fails to concentrate on the unit circle, or in the beta model that fails to match the predicted rotationally symmetric density.

Figures

Figures reproduced from arXiv: 2604.07647 by Arnab Sen, Ohad Noy Feldheim.

Figure 1
Figure 1. Figure 1: Roots of the uniform model for n = 1200. 1.2 The Beta Model While the uniform model is perhaps the most natural one to consider, Theorem 1 shows that its macroscopic root distribution lies in the same universality class as that of a Kac polynomial. The reason is that conditioning on log-concavity makes the coefficients vary only polynomially in n; indeed, most coefficients are of the same order. This place… view at source ↗
Figure 2
Figure 2. Figure 2: Left: roots of the beta model for n = 400 (orange) and n = 800 (blue), restricted to [−2, 2]2 . Right: histogram of log|ζn,k| for n = 800, overlaid with the theoretical log-radial density flog|ζ| (x) = 2 (4+|x|) 2 for x ∈ R. Although the limiting root distribution is rotationally symmetric, the visible cone around the positive x-axis that seems to contain no zeros is somewhat puzzling. Since all coefficien… view at source ↗
read the original abstract

We introduce two probabilistic models of random log-concave polynomials, the uniform model and the beta model, and study the asymptotic distribution of their zeros in the complex plane. In the uniform model, we show that the empirical root distribution converges to the uniform probability measure on the unit circle, placing the model in the same universality class as classical Kac polynomials. In contrast, in the beta model log-concavity is amplified through exponential scaling of the coefficients, leading to a new limiting distribution that is rotationally symmetric and absolutely continuous with respect to Lebesgue measure on the plane.

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

0 major / 2 minor

Summary. The manuscript introduces two probabilistic models for random log-concave polynomials—the uniform model and the beta model—and derives the limiting empirical distributions of their complex roots. In the uniform model the roots converge to the uniform probability measure on the unit circle, placing the model in the same universality class as classical Kac polynomials. In the beta model, exponential scaling of the coefficients produces a new rotationally symmetric limiting distribution that is absolutely continuous with respect to Lebesgue measure on the plane.

Significance. If the stated convergence theorems hold, the work is significant for extending the study of random polynomials to the log-concave coefficient regime. The uniform model recovers a known limiting law, while the beta model supplies an explicit new rotationally symmetric density; both results are falsifiable and could serve as benchmarks for further analytic or numerical investigations in random polynomials and complex analysis.

minor comments (2)
  1. The abstract and introduction should explicitly state the range of polynomial degrees n for which the limiting statements are proved and whether the results are uniform in the model parameters.
  2. Notation for the coefficient distributions (especially the precise form of the beta-model density) should be introduced once in a dedicated preliminary section and used consistently thereafter.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our manuscript and the recommendation of minor revision. The referee's description accurately reflects the two models introduced and the limiting root distributions obtained. No specific major comments or requested changes appear in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines the uniform and beta models explicitly as probabilistic constructions that generate log-concave coefficient sequences, then derives the limiting root distributions (uniform on the unit circle; rotationally symmetric absolutely continuous measure) as consequences of asymptotic analysis on those models. No quoted equations or steps reduce a claimed prediction back to a fitted input or self-definition by construction. No load-bearing self-citations or uniqueness theorems imported from prior author work appear in the provided claims. The universality-class placement follows directly from the stated convergence result rather than presupposition. The derivation remains self-contained against the model definitions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are identifiable from the abstract. The models are introduced as probabilistic constructions, but their precise definitions, any scaling parameters, or background assumptions are not stated.

pith-pipeline@v0.9.0 · 5380 in / 1162 out tokens · 60308 ms · 2026-05-10T16:51:41.320025+00:00 · methodology

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