On the Golomb-Dickman constant under Ewens sampling

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.