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Zeros and exponential profiles of polynomials II: Examples

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Pith's one-line read One exponential profile predicts zero limits of a dozen polynomial families

desk verdict A solid sequel that makes the exponential-profile method into a usable catalog, with a couple of genuinely new finite-free constructions. read the letter →

arxiv 2509.11248 v1 pith:HXJI472L submitted 2025-09-14 math.CA math.PR

Jonas Jalowy , Zakhar Kabluchko , Alexander Marynych This is my paper · ORCID
classification math.CAmath.PR MSC 26C1060B1033C4546L5430C1030C1560B2011B73
keywords exponentialprofileempiricalzerodistributionreal-rootedpolynomialsCauchytransformTouchardMarchenko-Pasturlawfreemultiplicativenormalsamplecovariancematrix
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to show that the limiting distribution of zeros of many well-known real-rooted polynomial families can be read off from one function: the exponential profile of their coefficients. If a sequence of polynomials has nonpositive roots and the normalized logarithms of its coefficients converge to a profile g, then the roots converge to a measure whose Cauchy transform is obtained by inverting the derivative of g. The authors apply this recipe to Stirling, Touchard, Fubini, Eulerian, Narayana, hypergeometric, Laguerre, Hermite, Jacobi, and little q-Laguerre polynomials, and they construct new polynomial families that converge to the free multiplicative normal and Poisson distributions. As an inverse application, they derive a law of large numbers for the coefficients of the characteristic polynomial of sample covariance matrices. A sympathetic reader would care because computing a profile is often easier than direct zero analysis, turning many scattered limit results into corollaries of one transfer theorem.

What carries the argument

The exponential profile g, defined as the limit of n^{-1} log a_{⌊αn⌋:n} over the degree index α, is the central object. The transfer theorem (Theorem 2.4) converts g into the limiting zero measure via the inverse-function relation tG(t) = (e^{−g′})^{-1}(t), where G is the Cauchy transform. For the free-probability constructions, Proposition 5.1 shows that if b_{k:n} has an exponential profile g̃, then the finite free convolution power has S-transform e^{g̃′(1+t)}; Laguerre's theorem is used to guarantee real-rootedness of the constructed polynomials.

What would settle it

Compute the roots for a moderately large n in a listed family, for example little q-Laguerre with q=e^{−λ/n} or Fubini polynomials, and compare the empirical zero measure with the predicted density; any systematic mismatch, or a single complex root in a family claimed to be real-rooted, would falsify that row of the table.

Watch

Extended reading notes

Core claim

The paper's central claim is that the exponential profile of a sequence of polynomials with nonpositive roots completely determines the limiting empirical zero distribution. Specifically, Theorem 2.4 (quoted from the authors' earlier work) says that if the k-th coefficient a_{k:n} satisfies log a_{⌊αn⌋:n}/n → g(α) for α in an interval, then the empirical zero measures converge, and the limiting Cauchy transform G satisfies that t↦tG(t) is the inverse of α↦e^{−g′(α)}. The paper's contribution is to verify the profile and nonpositive-root hypotheses for a wide table of classical families and to build new families from finite free convolution whose S-transforms are e^{g′(1+t)}, producing free m

Load-bearing premise

The whole table depends on the companion transfer theorem being valid and on each family genuinely having only nonpositive real roots; the paper often cites real-rootedness from other sources rather than proving it, so a single overlooked failure of real-rootedness would break that family's limit statement.

Editorial extensions

If this is right

  • The limits compiled in Table 1 become corollaries of one transfer result: Touchard polynomials converge to a measure with moments (−k)^k/(k+1)!, Fubini to a Cauchy-type density, Eulerian to log-Cauchy, Laguerre to Marchenko–Pastur, and Hermite to the semicircle law.
  • New polynomial families G_n(x;σ²/n) and P_n(x;b_n,c_n) give explicit finite-free-probability models whose zero limits are the free multiplicative normal and Poisson distributions.
  • The inverse direction yields coefficient asymptotics for characteristic polynomials of sample covariance matrices: log a_{k:n} ≈ n g_M(k/n) with an explicit profile g_M.
  • Little q-Laguerre polynomials with q=e^{−λ/n} connect to q-deformed GUE densities and recover Marchenko–Pastur as λ→0.
  • Any real-rooted polynomial sequence with nonpositive roots and a computable profile can be plugged into the same recipe, so the method applies beyond the families listed.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the transfer is robust, the paper suggests a general 'profile-first' strategy: real-rootedness is the main obstruction, and many combinatorial families with known coefficient asymptotics should have zero limits that are free or classical laws waiting to be identified.
  • The free multiplicative normal and Poisson constructions hint that finite free convolution powers can synthesize polynomials with prescribed free multiplicative limits; one might test whether fractional powers extend the semigroup and yield new infinite-divisibility results.
  • The covariance-matrix coefficient asymptotics could be repeated for other random matrix ensembles with known limiting spectral measures, giving coefficient laws of large numbers via the same converse theorem.
  • The q-Laguerre family interpolates between Marchenko–Pastur (λ→0) and q-deformed distributions; making that interpolation quantitative could yield new spectral statistics for q-Hermite ensembles.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 4 minor

Summary. The paper applies the exponential-profile method developed in the authors' companion paper [31] to a broad catalogue of polynomial families. For each family with nonpositive real roots and a coefficient profile, Theorem 2.4 converts the profile into the limiting empirical zero distribution via the inverse of a simple function. The catalogue covers Stirling, Touchard, Fubini, Eulerian, Narayana, hypergeometric, Laguerre, Hermite, Jacobi and little q-Laguerre polynomials. The paper also constructs new polynomial families whose zero distributions converge to the free multiplicative normal and Poisson distributions, treats differential-operator-generated families, and uses the converse theorem to obtain coefficient asymptotics for characteristic polynomials of sample covariance matrices. The resulting Table 1 is the main deliverable.

Significance. If the results hold, this is a useful unifying reference: a large collection of classical and free-probability examples are derived from one transfer principle, with explicit Cauchy transforms, densities, moments and logarithmic potentials. The new free multiplicative Hermite and Poisson polynomial constructions are of independent interest, and the inverse result in Section 6 gives a concrete new coefficient-limit theorem for random covariance matrices. The paper explicitly depends on the companion paper [31] for the central transfer theorem and on several imported real-rootedness results; that division of labour is normal for a sequel and not by itself a defect. The derivations I spot-checked (Touchard, Fubini, Eulerian, Laguerre/Marchenko–Pastur, Hermite/semicircle, and the hypergeometric rational profile) are algebraically consistent.

major comments (2)
  1. [§5.1.2, Lemma 5.9] The first proof for the case b≤-n is incorrect as written. For f(z)=(-b-z)^c, the only zero is z=-b, which is ≥ n > 0, not nonpositive. Hence f is not in the Laguerre–Pólya class with negative zeros and Laguerre's theorem (Theorem 5.6) cannot be applied. Remark 5.13 supplies a valid proof for b≠-n via the characteristic-polynomial identity, but the case b=-n is left unproved. Since Theorem 5.12 explicitly allows β=-1, this needs a fix, e.g. a continuity argument in b, or a separate treatment of b=-n.
  2. [§5.1.2, proof of Theorem 5.12] The proof applies Proposition 5.1 to Q_n with b_{k:n}=|(k+b_n)/n|^{c_n/n}. Proposition 5.1 requires Q_n to be real-rooted, but this is not verified and is in general false for non-integer exponents (cf. Remark 5.10). The conclusion can be obtained directly from the exponential profile of P_n, whose real-rootedness is the content of Lemma 5.9, together with [31, Lemma 6.1] for the S-transform. The gap is in the proof rather than in the statement, but it should be repaired.
minor comments (4)
  1. [§5.1, Proposition 5.1 and Remark 5.13] The displayed equations in these passages contain long repeated garbled strings ('⌟⟨⟨⟪rl⟫l⟩⟩⟪⌟⟪...') that make the definitions almost unreadable. This appears to be an encoding/compilation error and should be cleaned up before publication.
  2. [§4.7, text after Corollary 4.14] The statement 'One can check that t↦tG(t) is a solution to the quadratic equation' is slightly compressed; giving the explicit coefficient (σ^2 λ α^2 + (t+σ^2(1-λ))α - t = 0) in the prose would help the reader verify the inversion.
  3. [§4.8, Remark 4.16] The sentence 'the semicircle limiting distribution for Hen could have been derived directly ... using a trick discussed in Remark 2.6' is vague. A one-sentence indication of the shift (e.g. A = n^{1/2} sup |zeros|) would be useful.
  4. [§4.10, equation (47)] In the density formula, the notation p_{a|λ}(x) is used for a measure on [0,∞), while the coefficient profile was computed for P_n(-x). The reflection convention is clear from context, but a short sentence would prevent confusion.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the general profile-to-zero transfer is a companion-paper theorem independent of the examples; all applications use independent coefficient asymptotics and real-rootedness inputs.

full rationale

I walked the derivation chain. The key transfer results (Theorem 2.4, Theorem 2.5, and Lemma 6.1 of [31]) are cited from the authors' companion paper, but they are general, parameter-free statements about exponential profiles and Cauchy transforms; their statements do not mention any of the specific polynomial families or target measures, so using them is a normal division of labor rather than a circular reduction. Every application computes the exponential profile independently from classical asymptotic estimates (Stirling asymptotics, Bender's formulas, Riemann-sum limits) and then inverts alpha |-> exp(-g'(alpha)) exactly as Theorem 2.4 prescribes. The real-rootedness hypotheses are imported from external sources (Harper, Mez<0x0151>, [42], [52], Laguerre's theorem) or proved in the paper; none of these inputs equals the claimed zero-density conclusions. The new free-probability constructions are explicit coefficient choices whose limits are derived from the profile, not fitted to the stated limits. Section 6 applies the converse Theorem 2.5 to a known Marchenko-Pastur limit to obtain coefficient asymptotics, which is an inverse result, not a prediction from fitted parameters. The only real defect I found, Lemma 5.9's assertion that (-b-z)^c has nonpositive roots for b <= -n when the actual root is z = -b > 0, is a proof gap rather than circularity, and the paper supplies an alternative argument in Remark 5.13 covering the required cases. Thus no load-bearing step reduces to its own inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted to data; constants such as sigma, gamma, lambda, q, a, b index the families and are treated as inputs. No new physical entities are postulated. The new polynomial families are explicit mathematical constructions, not entities with independent falsifiable handles. The main burden on the reader is accepting the companion theorems and the cited real-rootedness results.

assumptions (5)
  • standard math Theorem 2.4 and Theorem 2.5 from companion paper [31]: exponential profile of coefficients is equivalent to weak convergence of empirical zero distributions, with the Cauchy transform recovered from e^{-g'(alpha)}.
    Invoked throughout, e.g., in every proof in Section 4 and in Proposition 5.1. The proof is not reproduced in this paper but is delegated to [31].
  • domain assumption Real-rootedness (with nonpositive roots) of the polynomial families under consideration.
    Needed to apply Theorem 2.4. For Touchard this is Harper [30]; for Fubini [47]; for Narayana [37]; for hypergeometric polynomials [42, Theorems 4.6 and 4.7]; for q-Laguerre [35]; for Laguerre-Hermite relations [60].
  • standard math Laguerre's theorem: if f is in the Laguerre-Polya class and p has all real roots, then the coefficientwise transformed polynomial has all real roots.
    Used in the proofs of Theorem 5.3 and Lemma 5.9 to establish real-rootedness of the new free-probability polynomial families.
  • standard math Classical logarithmic asymptotics for Stirling numbers and factorials, e.g., Bender [7], Moser-Wyman [49].
    The exponential profiles for Stirling, Touchard, Fubini, and Eulerian polynomials are derived from these asymptotic formulas.
  • standard math Marchenko-Pastur theorem for sample covariance matrices, e.g., Bai-Silverstein [5, Theorem 3.6].
    Section 6 uses the almost sure weak convergence of the empirical eigenvalue distribution of sample covariance matrices as the input for the converse Theorem 2.5.

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Pith. "Pith review of Zeros and exponential profiles of polynomials II: Examples." pith.science (2026). https://pith.science/paper/HXJI472L

@misc{pith2026250911248,
  author       = {Pith},
  title        = {Pith review of: Zeros and exponential profiles of polynomials II: Examples},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HXJI472L}},
  note         = {Machine review of arXiv:2509.11248}
}
abstract

In [Jalowy, Kabluchko, Marynych, arXiv:2504.11593v1, 2025], the authors discuss a user-friendly approach to determine the limiting empirical zero distribution of a sequence of real-rooted polynomials, as the degree goes to $\infty$. In this note, we aim to apply it to a vast range of examples of polynomials providing a unifying source for limiting empirical zero distributions. We cover Touchard, Fubini, Eulerian, Narayana and little $q$-Laguerre polynomials as well as hypergeometric polynomials including the classical Hermite, Laguerre and Jacobi polynomials. We construct polynomials whose empirical zero distributions converge to the free multiplicative normal and Poisson distributions. Furthermore, we study polynomials generated by some differential operators. As one inverse result, we derive coefficient asymptotics of the characteristic polynomial of random covariance matrices.

Figures

Figures reproduced from arXiv: 2509.11248 by the authors.

Figure 1
Figure 1. The limiting probability densities of zeros for generating polynomials of Stirling numbers of the first kind (left), Touchard polynomials (center) and Fubini polynomials (right). Inverting this function we obtain GT (t) = t −1w ← T (W0(1/t)) = 1 − e −W0(1/t) tW0(1/t) = e W0(1/t) − 1 tW0(1/t)eW0(1/t) = e W0(1/t) − 1. (19) Formula (15) for the density pT follows by the Stieltjes–Perron inversion (4). To prove formula … view at source ↗
Figure 2
Figure 2. The limiting probability densities of zeros of Eulerian polynomials (left), Narayana polynomials for γ = 2 (center) and γ = 10 (right). 4.5. Narayana polynomials. The Narayana numbers and the corresponding polynomials are defined by Nn,k = 1 k + 1 ( n k )(n − 1 k ), k ∈ {0, . . . , n − 1}, Nn(x) = n ∑ k=1 Nn,k−1x k , see [37, Chapter 2] for their definition and properties. It is known [37, p. 90, Exercise 4.7] that … view at source ↗
Figure 3
Figure 3. The limiting probability densities of zeros of Laguerre polynomials for γ ∗ ∈ {0, 1, 5} (from left to right) and Hermite polynomials (right). 4.8. Hermite polynomials. The classical (probabilist) Hermite polynomials are given by Hen(x) = e − 1 2 D2 x n = n! ⌊n/2⌋ ∑ m=0 (−1) m m!2m ⋅ x n−2m (n − 2m)! , (37) where we used the notation D = d dx . It is known that Hermite polynomials have only real roots and these roots… view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: The limiting distribution of zeros of Legendre polynomials (left, γ = 1/2), Gegenbauer polynomials for γ = 1 (center) and for γ = 5 (right). The case u = v = 0 corresponds to the Legendre polynomials with Gµ J (0,0) (t) = 2 √ t 2 − 1 , t ∉ [−1, 1], (44) which means tha…
Figure 5
Figure 5. Figure 5: The limiting probability densities of zeros of little q-Laguerre polynomi￾als for a = 1 in the top row with λ ∈ {1/10, log(2), 2} (from left to right) and in the bottom row for a = 1/3, λ = 3 (left), a = 1/2, λ = 1/4 (center) and a = 1/100, λ = 1/5 (right). of the loga…

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Pith tools

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