Poisson–Dirichlet conjecture for macroscopic loop lengths

Let $\ell_1,\ell_2,\dots$ be the loop lengths ordered decreasingly, let $m(\beta)$ be the long-loop mass, and let $u$ be the model parameter. Define

$$\vartheta=\begin{cases}1,&u=0\text{ or }u=1,\frac12,&u\in(0,1). \end{cases}$$

Write $\operatorname{PD}_{[0,m(\beta)]}(\vartheta)$ for the distribution obtained by multiplying a Poisson–Dirichlet partition with parameter $\vartheta$ by $m(\beta)$. Macroscopic-loop Poisson–Dirichlet conjecture. For every $k$, the joint distribution of $\ell_1,\dots,\ell_k$ converges to the joint distribution of the first $k$ elements of a random partition with distribution $\operatorname{PD}_{[0,m(\beta)]}(\vartheta)$. This is the paper's proposed universal law for macroscopic loops; the supplied material does not establish it.