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On the Golomb-Dickman constant under Ewens sampling

José Ricardo G. Mendonça, Luis Jehiel Negret

Abstract

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 $θ$, and illustrate our results with numerical computations and Monte Carlo simulations of the Hoppe urn. Our results can be viewed as an extension of the classical calculations of Shepp and Lloyd to the Ewens setting by relatively elementary means.

On the Golomb-Dickman constant under Ewens sampling

Abstract

We define a generalized Golomb-Dickman constant as the limiting expected proportion of the longest cycle in random permutations under the Ewens measure with parameter . Exploiting the independence properties of Kingman's Poisson process construction of the Poisson-Dirichlet distribution, we obtain an explicit integral representation for in terms of the exponential integral. The dependence of on reflects the transition between regimes dominated by long cycles (small ) and those with many small cycles (large ). We also derive the asymptotic behavior of for small and large , and illustrate our results with numerical computations and Monte Carlo simulations of the Hoppe urn. Our results can be viewed as an extension of the classical calculations of Shepp and Lloyd to the Ewens setting by relatively elementary means.
Paper Structure (9 sections, 4 theorems, 28 equations, 3 figures, 1 table)

This paper contains 9 sections, 4 theorems, 28 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The Poisson representation of Ewens permutations
  3. The scaling limit of the largest Poisson atom
  4. Integral representation of $\lambda_{\theta}$ and its asymptotics
  5. The main theorem
  6. Asymptotic behavior of $\lambda_{\theta}$
  7. Numerics, simulations and applications
  8. Numerical values
  9. Simulation of the Hoppe urn model

Key Result

Lemma 3.1

For each $k \ge 1$, we have Moreover, under the scaling $x = sk$, we have where is the exponential integral function of a nonnegative real argument Jeffreys1972.

Figures (3)

  • Figure 1: Generalized Golomb--Dickman constant $\lambda_{\theta}$, equation \ref{['eq:ggd']}.
  • Figure 2: Timeline of the color proportions in one simulation of $50$ draws in a Hoppe urn with $\theta=1$.
  • Figure 3: Monte Carlo estimates of $\lambda_{\theta}$ obtained via the Hoppe urn simulation compared with the theoretical curves.

Theorems & Definitions (13)

  • Remark 2.1
  • Lemma 3.1
  • proof
  • Theorem 4.1
  • proof
  • Remark 4.2
  • Remark 4.3
  • Proposition 4.4
  • proof
  • Proposition 4.5
  • ...and 3 more