On the Deepest Cycle of a Random Mapping

TL;DR

This paper investigates the properties of the deepest cycle in a random mapping, establishing its distributional convergence, expected size, and the proportion of cyclic vertices involved, revealing insights into the structure of random mappings.

Contribution

It introduces the asymptotic distribution of the deepest cycle length in random mappings and quantifies the likelihood of vertices belonging to this cycle versus the largest component.

Findings

Distribution of cycle length converges when scaled by √n.

Approximately 55% of cyclic vertices are in the deepest cycle.

About 7.5% of vertices on the longest cycle are outside the largest component.

Abstract

Let $\mathcal{T}_n$ be the set of all mappings $T:\{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$. The corresponding graph of $T$ is a union of disjoint connected unicyclic components. We assume that each $T\in\mathcal{T}_n$ is chosen uniformly at random (i.e., with probability $n^{-n}$). The cycle of $T$ contained within its largest component is callled the deepest one. For any $T\in\mathcal{T}_n$, let $\nu_n=\nu_n(T)$ denote the length of this cycle. In this paper, we establish the convergence in distribution of $\nu_n/\sqrt{n}$ and find the limits of its expectation and variance as $n\to\infty$. For $n$ large enough, we also show that nearly $55\%$ of all cyclic vertices of a random mapping $T\in\mathcal{T}_n$ lie in the deepest cycle and that a vertex from the longest cycle of $T$ does not belong to its largest component with approximate probability $0.075$.