Poisson–Dirichlet law conjecture for normalized cycle lengths

Under the preceding critical-temperature conjecture, let $\xi\in\Delta_1$ be the weak limit of the ordered normalized cycle-length vector. Poisson–Dirichlet law conjecture. The distribution of $\xi$ is $\mathrm{PD}_{\theta}$ for an appropriate choice of $\theta$. This refines the prediction that the model has many macroscopic cycles rather than one giant cycle; the parameter $\theta$ is expected to be determined by the effective split-merge rates.