Parking functions are combinatorial objects that lie between permutations and mappings. We study the structure of cycles in the digraph representation of uniformly random parking functions. This has been a challenging research topic, as random parking functions do not satisfy nice exchangeability properties like random permutations and random mappings do.

Our results are multifold and demonstrate the asymptotic equivalence of ensembles between parking functions and mappings: we obtain an explicit formula for the number of parking functions with a prescribed number of cyclic points and show that the scaled number of cyclic points of a random parking function is asymptotically Rayleigh distributed; we establish the classical trio of limit theorems (law of large numbers, central limit theorem, large deviation principle) for the number of cycles in a random parking function; we also compute the asymptotic mean of the length of the rth longest cycle in a random parking function for all valid r.

A variety of tools from probability theory and combinatorics are used in our investigation. Joint work with J. E. Paguyo.