Poisson–Dirichlet limit for normalized largest component sizes

Prove that the process of normalized largest component sizes of a uniformly random parking function, namely (\tilde{L}_1(\pi_n)/n, \tilde{L}_2(\pi_n)/n, \ldots), converges in distribution to the Poisson–Dirichlet distribution with parameter 1/2.

Background

The paper establishes results for cycle lengths and observes that, after conditioning on the number of cyclic points, the normalized cycle-length process converges to the Poisson–Dirichlet distribution with parameter 1. The unresolved issue concerns component sizes rather than cycle lengths. Here, \tilde{L}_r(\pi_n) denotes the size of the rth largest connected component of the digraph associated with a uniformly random parking function.

For random mappings, Aldous proved convergence of normalized largest component sizes to the Poisson–Dirichlet distribution with parameter 1/2. The authors conjecturally anticipate the same limiting behavior for random parking functions, but explicitly state that a rigorous proof requires a deeper understanding of their component structure.