On the Golomb-Dickman constant under Ewens sampling

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, Monte Carlo simulations of the Hoppe urn, and an application.