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Limit distributions for cycles of random parking functions

J. E. Paguyo, Mei Yin

TL;DR

This paper analyzes the asymptotic cycle structure of uniformly random parking functions by embedding them into mapping/digraph models and establishing an equivalence of ensembles with random mappings. It provides explicit enumerative formulas for the number of parking functions with a fixed number of cyclic points, proves that the scaled number of cyclic points converges to a Rayleigh(1) distribution, and derives the classical trio of limit theorems (law of large numbers, central limit theorem, large deviation principle) for the number of cycles, including the asymptotic mean of the r-th longest cycle. The results extend to prime parking functions with identical asymptotics, offering a comprehensive probabilistic picture of cyclic structure and longest cycles in this combinatorial model. These findings illuminate the connections between parking functions, mappings, and Poisson–Dirichlet-type limits, and open avenues for deeper analysis of tree components and connected components within parking function digraphs.

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.

Limit distributions for cycles of random parking functions

TL;DR

This paper analyzes the asymptotic cycle structure of uniformly random parking functions by embedding them into mapping/digraph models and establishing an equivalence of ensembles with random mappings. It provides explicit enumerative formulas for the number of parking functions with a fixed number of cyclic points, proves that the scaled number of cyclic points converges to a Rayleigh(1) distribution, and derives the classical trio of limit theorems (law of large numbers, central limit theorem, large deviation principle) for the number of cycles, including the asymptotic mean of the r-th longest cycle. The results extend to prime parking functions with identical asymptotics, offering a comprehensive probabilistic picture of cyclic structure and longest cycles in this combinatorial model. These findings illuminate the connections between parking functions, mappings, and Poisson–Dirichlet-type limits, and open avenues for deeper analysis of tree components and connected components within parking function digraphs.

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 th longest cycle in a random parking function for all valid . 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.

Paper Structure

This paper contains 11 sections, 19 theorems, 57 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Literature review
  3. Our contributions
  4. Preliminaries
  5. Notations and definitions
  6. Auxiliary results
  7. Distribution of cyclic points
  8. Trio of limit theorems for the number of cycles
  9. Lengths of the longest cycles
  10. Limit theorems for prime parking functions
  11. Final remarks

Key Result

Proposition 2.1

Let $K_n(\sigma_n)$ be the number of cycles of a uniformly random permutation $\sigma_n \in S_n$. Then the mean and variance of $K_n(\sigma_n)$ are respectively given by Moreover, asymptotically we have as $n \to \infty$.

Figures (2)

  • Figure 1: The digraph representation of a uniformly random parking function of length 20.
  • Figure 2: The digraph representations of parking functions $\pi^1_{20}$ (top plot) and $\pi^2_{20}$ (middle plot) and non-parking function $\pi^3_{20}$ (bottom plot). Both $\pi^2_{20}$ and $\pi^3_{20}$ are transformed from $\pi^1_{20}$ by permuting vertices.

Theorems & Definitions (36)

  • Proposition 2.1: Gon44
  • Proposition 2.2: SL66
  • Proposition 2.3: KT76
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • proof
  • Proposition 3.4
  • ...and 26 more