Limit Theorems for Patterns in Ranked Tree-Child Networks

TL;DR

This paper establishes limit laws for the frequency of specific patterns in randomly chosen ranked tree-child networks, extending previous results and classifying their asymptotic behaviors.

Contribution

It extends the limit law for cherries to a broader class of patterns and classifies their asymptotic occurrence in ranked tree-child networks.

Findings

Patterns of height 1 and 2 occur either frequently, sporadically, or not at all.

The limit laws are normal, Poisson, or degenerate depending on the pattern.

The results suggest these are the only possible limit laws for fringe patterns.

Abstract

We prove limit laws for the number of occurrences of a pattern on the fringe of a ranked tree-child network which is picked uniformly at random. Our results extend the limit law for cherries proved by Bienvenu et al. (2022). For patterns of height $1$ and $2$, we show that they either occur frequently (mean is asymptotically linear and limit law is normal) or sporadically (mean is asymptotically constant and limit law is Poisson) or not all (mean tends to $0$ and limit law is degenerate). We expect that these are the only possible limit laws for any fringe pattern.