Large Components and Trees of Random Mappings

TL;DR

This paper analyzes the structure of random mappings by studying the sizes of components and trees within their functional digraphs, providing asymptotic probabilities for a vertex belonging to specific largest trees.

Contribution

It computes the limiting conditional probability that a vertex in the largest component belongs to its s-th largest tree, addressing a problem posed by Mutafchiev and Finch (2024).

Findings

Derived the asymptotic probability for a vertex to be in the s-th largest tree.

Provided insights into the structure of large components in random mappings.

Addressed a recent open problem in the study of random functional graphs.

Abstract

Let $\mathcal{T}_n$ be the set of all mappings $T:[n]\to[n]$, where $[n]=\{1,2,\ldots,n\}$. The corresponding graph $G_T$ of $T$, called a functional digraph, is a union of disjoint connected components. Each component is a directed cycle of rooted labeled trees. We assume that each $T\in\mathcal{T}_n$ is chosen uniformly at random from the set $\mathcal{T}_n$. The components and trees of $G_T$ are distinguished by their size. In this paper, we compute the limiting conditional probability ($n\to\infty$) that a vertex from the largest component of the random graph $G_T$, chosen uniformly at random from $[n]$, belongs to its $s$-th largest tree, where $s\ge 1$ is a fixed integer. This limit can be also viewed as an approximation of the probability that the $s$-th largest tree of $G_T$ is a subgraph of its largest component, which is a solution of a problem suggested by Mutafchiev and Finch (2024).