On the Golomb-Dickman constant under Ewens sampling

TL;DR

This paper introduces a generalized Golomb-Dickman constant under Ewens measure, providing explicit formulas, asymptotic analysis, and numerical validation for the expected proportion of the longest cycle in random permutations.

Contribution

It derives an explicit integral representation for the generalized constant and analyzes its dependence on the Ewens parameter, connecting cycle structure regimes.

Findings

Explicit integral formula for $mbda_ heta$ in terms of exponential integral.

Asymptotic behavior of $mbda_ heta$ for small and large $ heta$.

Numerical and simulation results illustrating the theoretical findings.

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.