Tree and connected component asymptotics for random parking functions

Determine the asymptotic behavior of the tree components and connected components of uniformly random parking functions, and rigorously establish whether the equivalence of ensembles with random mappings holds for these characteristics.

Background

The paper develops limit theorems for cycle-related statistics of uniformly random classical and prime parking functions and shows that several such statistics exhibit asymptotic equivalence with corresponding statistics for random mappings. However, the structural parts of the associated functional digraphs that are attached to cycles are not analyzed: these include the tree components and the connected components themselves.

The authors explicitly leave their asymptotic investigation unresolved and note that proving equivalence of ensembles for these characteristics is expected to be difficult. Resolving this problem would extend the paper’s cycle-level results to the broader component structure of random parking functions.