Limit distributions for cycles of random parking functions

TL;DR

This paper investigates the asymptotic properties of cycles in random parking functions, providing explicit formulas, distributional limits, and asymptotic cycle length estimates using probabilistic and combinatorial methods.

Contribution

It introduces explicit formulas for the number of parking functions with a given number of cycles and establishes their asymptotic distributional behavior, including Rayleigh distribution and classical limit theorems.

Findings

Number of cyclic points follows Rayleigh distribution asymptotically

Established law of large numbers, CLT, and large deviations for cycle counts

Computed asymptotic mean length of the rth longest cycle

Abstract

We study the asymptotic behavior of cycles of uniformly random parking functions. Our results are multifold: 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 $r$th 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. Corresponding results for the class of prime parking functions are obtained.