Components and Cycles of Random Mappings

TL;DR

This paper investigates the structure of random mappings, deriving new formulas for cycle lengths and analyzing the effects of constraints on component structures, with precise asymptotic results.

Contribution

It introduces a new density formula for the longest cycle length in random mappings and explores the impact of the single-component constraint.

Findings

Mean length of the longest cycle is approximately 0.7824 times sqrt(n).

Under the single-component constraint, the mean length is about 0.7978 times sqrt(n).

The difference between these means is less than 2%.

Abstract

Each connected component of a mapping $\{1,2,...,n\}\rightarrow\{1,2,...,n\}$ contains a unique cycle. The largest such component can be studied probabilistically via either a delay differential equation or an inverse Laplace transform. The longest such cycle likewise admits two approaches: we find an (apparently new) density formula for its length. Implications of a constraint -- that exactly one component exists -- are also examined. For instance, the mean length of the longest cycle is $(0.7824...)\sqrt n$ in general, but for the special case, it is $(0.7978...)\sqrt n$, a difference of less than $2\%$.