On the Golomb-Dickman constant under Ewens sampling
Fuente:
arXiv
Enregistré dans:
| Auteurs principaux: | , |
|---|---|
| Format: | Preprint |
| Publié: |
2026
|
| Sujets: | |
| Accès en ligne: | |
| Tags: |
Ajouter un tag
Pas de tags, Soyez le premier à ajouter un tag!
|
| _version_ | 1866914586688487424 |
|---|---|
| author | Mendonça, José Ricardo G. Negret, Luis Jehiel |
| author_facet | Mendonça, José Ricardo G. Negret, Luis Jehiel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_23175 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the Golomb-Dickman constant under Ewens sampling Mendonça, José Ricardo G. Negret, Luis Jehiel Probability Statistical Mechanics Statistics Theory Primary: 60C05, Secondary: 05A05, 60F05, 60G55 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. |
| title | On the Golomb-Dickman constant under Ewens sampling |
| topic | Probability Statistical Mechanics Statistics Theory Primary: 60C05, Secondary: 05A05, 60F05, 60G55 |
| url | https://arxiv.org/abs/2603.23175 |