Cycle structure of random parking functions

TL;DR

This paper investigates the cycle structure of random parking functions, deriving asymptotic expectations and demonstrating convergence to independent Poisson distributions using advanced combinatorial and probabilistic methods.

Contribution

It introduces the first detailed analysis of cycle counts in random parking functions, including asymptotic expectations and distributional convergence results.

Findings

Expected number of cycles of fixed length computed asymptotically

Total variation distance bounds between cycle counts and Poisson variables

Cycle counts converge to independent Poisson distributions under mild conditions

Abstract

We initiate the study of the cycle structure of uniformly random parking functions. Using the combinatorics of parking completions, we compute the asymptotic expected value of the number of cycles of any fixed length. We obtain an upper bound on the total variation distance between the joint distribution of cycle counts and independent Poisson random variables using a multivariate version of Stein's method via exchangeable pairs. Under a mild condition, the process of cycle counts converges in distribution to a process of independent Poisson random variables.