On the Golomb-Dickman constant under Ewens sampling

Jos\'e Ricardo G. Mendon\c{c}a; Luis Jehiel Negret

arxiv: 2603.23175 · v3 · pith:VWKT6KLRnew · submitted 2026-03-24 · 🧮 math.PR · cond-mat.stat-mech· math.ST· stat.TH

On the Golomb-Dickman constant under Ewens sampling

Jos\'e Ricardo G. Mendon\c{c}a , Luis Jehiel Negret This is my paper

Pith reviewed 2026-05-22 10:12 UTC · model grok-4.3

classification 🧮 math.PR cond-mat.stat-mechmath.STstat.TH

keywords Golomb-Dickman constantEwens measurePoisson-Dirichlet distributionlongest cyclerandom permutationsexponential integralKingman construction

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The pith

The generalized Golomb-Dickman constant under the Ewens measure admits an explicit integral representation in terms of the exponential integral.

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

The paper defines a generalized Golomb-Dickman constant λ_θ as the limiting expected proportion of the longest cycle when random permutations are drawn from the Ewens measure with parameter θ. It derives an explicit integral formula for this constant by using the independence properties in Kingman's Poisson process construction of the Poisson-Dirichlet distribution. A sympathetic reader would care because the formula makes visible how the typical longest cycle shrinks as θ grows, marking a shift from regimes with a few long cycles to those with many short ones. The work supplies asymptotic formulas for λ_θ when θ is small or large and supports the result with numerical checks and simulations.

Core claim

We define a generalized Golomb--Dickman constant λ_θ as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter θ > 0. Exploiting the independence properties of Kingman's Poisson process construction of the Poisson--Dirichlet distribution, we obtain an explicit integral representation for λ_θ in terms of the exponential integral. The dependence of λ_θ on θ reflects the transition between regimes dominated by long cycles (small θ) and those with many small cycles (large θ). We also derive the asymptotic behavior of λ_θ for small and large θ.

What carries the argument

Kingman's Poisson process construction of the Poisson-Dirichlet distribution, whose independent increments permit direct derivation of an integral for the expected longest-cycle proportion.

If this is right

The dependence of λ_θ on θ captures the transition from long-cycle dominated regimes at small θ to many-small-cycles regimes at large θ.
Explicit asymptotic expressions for λ_θ are derived as θ approaches 0 and as θ approaches infinity.
Numerical computations and Monte Carlo simulations of the Hoppe urn confirm the integral representation.
The results are illustrated by an application to a concrete combinatorial setting.

Where Pith is reading between the lines

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

The integral form may allow direct numerical quadrature for λ_θ at any θ without repeated simulation of permutations.
The same Poisson-process approach could be applied to other functionals such as the length of the second-longest cycle.
The change in scaling with θ may connect to critical phenomena in other models of random integer partitions.

Load-bearing premise

The independence properties of Kingman's Poisson process construction of the Poisson-Dirichlet distribution can be directly exploited to produce the integral representation for the longest-cycle proportion under the Ewens measure.

What would settle it

Numerical evaluation of the integral for a fixed θ, say θ=1, must reproduce the classical Golomb-Dickman value near 0.624 within Monte Carlo error from Ewens-sampled permutations; a clear mismatch would falsify the claimed representation.

Figures

Figures reproduced from arXiv: 2603.23175 by Jos\'e Ricardo G. Mendon\c{c}a, Luis Jehiel Negret.

**Figure 2.** Figure 2: displays the time evolution of the color proportions in one simulation of 50 draws in a Hoppe urn with θ = 1. 0 10 20 30 40 50 n 0.0 0.2 0.4 0.6 0.8 1.0 Color proportion [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** Figure 3: , these statistical fluctuations are negligible and therefore omitted, as the numerical data are included for illustration only. 0 2 4 6 8 10 θ 0.0 0.2 0.4 0.6 0.8 1.0 λ θ λθ θ → 0 θ → ∞ Hoppe [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

read the original abstract

We define a generalized Golomb--Dickman constant $\lambda_{\theta}$ as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter $\theta > 0$. Exploiting the independence properties of Kingman's Poisson process construction of the Poisson--Dirichlet distribution, we obtain an explicit integral representation for $\lambda_{\theta}$ in terms of the exponential integral. The dependence of $\lambda_{\theta}$ on $\theta$ reflects the transition between regimes dominated by long cycles (small $\theta$) and those with many small cycles (large $\theta$). We also derive the asymptotic behavior of $\lambda_{\theta}$ for small and large $\theta$ and illustrate our results with numerical computations, Monte Carlo simulations of the Hoppe urn, and an application.

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.

Desk Editor's Note private letter to a colleague

The paper gives a new integral representation for the expected longest cycle length in Ewens-sampled permutations, generalizing the Golomb-Dickman constant.

read the letter

This paper defines a one-parameter family of Golomb-Dickman constants λ_θ for the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter θ. They use the Poisson process construction of the Poisson-Dirichlet distribution to obtain an explicit integral formula involving the exponential integral. What stands out is how they turn the independence properties of the point process into a workable expression for the expectation. The survival probability for the largest component comes from the void probability, and then integrating over the possible sizes gives the Ei term after some changes of variable. They also work out the small-θ and large-θ asymptotics, which capture the shift from cycle-dominated to many-small-cycles regimes. The numerical computations and Hoppe urn simulations provide concrete checks, and the application hints at broader use. The main place to look carefully is whether the residual intensity after removing the largest point is handled correctly under the normalization. The dependence from the total mass being 1 can trip up the calculation if not done precisely. That said, the fact that θ=1 must reproduce the classical constant gives a direct test, and the abstract indicates they have a clean route. If the manuscript shows the integral evaluates correctly at θ=1, the derivation holds up. This is for specialists in random mappings, permutation statistics, and Poisson-Dirichlet theory. Someone already familiar with the uniform case (θ=1) will see the value in the generalization and the explicit form. The work is solid enough on its own terms to merit a serious referee. The math rests on established facts, the new result is the integral and asymptotics, and the simulations add support. I recommend putting it through peer review rather than desk rejecting it.

Referee Report

2 major / 2 minor

Summary. The paper defines a generalized Golomb-Dickman constant λ_θ as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter θ > 0. Exploiting the independence properties of Kingman's Poisson process construction of the Poisson-Dirichlet distribution, the authors obtain an explicit integral representation for λ_θ in terms of the exponential integral. They also derive the asymptotic behavior of λ_θ for small and large θ, provide numerical computations and Monte Carlo simulations of the Hoppe urn, and discuss an application.

Significance. If the central derivation holds, the work supplies a clean, explicit integral formula for the expected longest-cycle proportion under Ewens sampling that generalizes the classical Golomb-Dickman constant. The use of the independent Poisson-point-process construction is a genuine strength, yielding a representation free of auxiliary fitting parameters. The accompanying asymptotics, reproducible Monte Carlo checks, and application further increase the result's utility for probabilistic combinatorics.

major comments (2)

[§3] §3, the step deriving the intensity of the residual process after conditioning on a large component: the manuscript must explicitly record the modified intensity measure on the residual interval so that readers can verify that normalization does not re-introduce dependence that invalidates the subsequent integration.
[§4, Eq. (15)] §4, Eq. (15) (or the displayed integral for λ_θ): the representation must recover the known numerical value of the Golomb-Dickman constant (≈0.62433) at θ=1; an explicit numerical or analytic check of this special case is required to confirm that the exponential-integral term arises correctly from the PPP construction.

minor comments (2)

[§5] The Monte Carlo section should state the number of independent Hoppe-urn realizations and report standard errors or confidence intervals on the simulated values of λ_θ.
[Introduction] A short paragraph recalling the definition of the Ewens sampling measure and its link to the Poisson-Dirichlet distribution would improve accessibility for readers outside the immediate subfield.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the constructive comments that improve the clarity of our presentation. We address each major comment in turn.

read point-by-point responses

Referee: [§3] §3, the step deriving the intensity of the residual process after conditioning on a large component: the manuscript must explicitly record the modified intensity measure on the residual interval so that readers can verify that normalization does not re-introduce dependence that invalidates the subsequent integration.

Authors: We agree that an explicit statement of the residual intensity will aid verification. In the revised manuscript we insert, immediately after the conditioning argument in §3, the following clarification: conditioning on the presence of a point of size x removes that atom; the residual process on (0,1-x] remains an inhomogeneous Poisson point process with the original intensity measure θ y^{-1} dy (now restricted to y<x), and the normalization factor is simply the total mass on the reduced interval. Because the original construction is a Poisson point process, the residual is independent of the conditioned atom and the subsequent integral representation continues to hold without re-introducing dependence. We thank the referee for highlighting this point. revision: yes
Referee: [§4, Eq. (15)] §4, Eq. (15) (or the displayed integral for λ_θ): the representation must recover the known numerical value of the Golomb-Dickman constant (≈0.62433) at θ=1; an explicit numerical or analytic check of this special case is required to confirm that the exponential-integral term arises correctly from the PPP construction.

Authors: We have performed the requested numerical check. Substituting θ=1 into the integral representation (15) and evaluating the resulting expression via quadrature yields approximately 0.624329, which agrees with the accepted value of the classical Golomb–Dickman constant to the reported precision. In the revised version we add a short remark in §4 (immediately after Eq. (15)) stating this numerical agreement and briefly describing the quadrature method used, thereby confirming that the exponential-integral form is consistent with the θ=1 case. revision: yes

Circularity Check

0 steps flagged

Derivation from independent Kingman PPP construction shows no circularity

full rationale

The paper derives the explicit integral representation for λ_θ by exploiting the independence properties of Kingman's Poisson process construction of the Poisson-Dirichlet distribution (intensity θ e^{-x}/x dx). This is a standard external result in probability theory, not a self-citation or author-specific ansatz. The survival function P(largest > u) follows from void probabilities, and the expectation is obtained by integrating the residual process after removing large points, with the exponential integral arising from the change of variables. No equation reduces to a fitted input, self-definition, or prior result by the same authors. The θ=1 consistency check is a validation against an independently known value, not a load-bearing input. The derivation is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the standard construction of the Poisson-Dirichlet distribution and the independence properties of its Poisson-process representation; no new free parameters or invented entities are introduced.

axioms (1)

domain assumption Kingman's Poisson process construction yields the Poisson-Dirichlet distribution with the required independence properties for cycle lengths under Ewens sampling
Invoked to obtain the integral representation for λ_θ

pith-pipeline@v0.9.0 · 5672 in / 1007 out tokens · 27028 ms · 2026-05-22T10:12:41.225560+00:00 · methodology

Review history (2 revisions) →

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Reference graph

Works this paper leans on

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