Ancestral diversity in fragmentation trees

TL;DR

This paper studies ancestral diversity in fragmentation trees, analyzing the asymptotic behavior of the number of distinct ancestors for multiple groups, revealing phase transition phenomena across various tree models.

Contribution

It extends previous work by considering general values of k and a broad class of fragmentation trees, including stable Lévy and phylogenetic models.

Findings

Identifies asymptotic regimes for ancestral diversity measures.

Discovers phase transition phenomena depending on model parameters.

Generalizes ancestral diversity analysis beyond Brownian trees.

Abstract

In a deterministic or random tree, a notion of ancestral diversity can be defined as follows. Sample independently $n$ groups of $k$ leaves and count the number $N_n(k)$ of distinct most recent common ancestors of each of the groups. As $n$ becomes large, the asymptotic behavior of $N_n(k)$ depends of course on the structure of the tree. Motivated by the study of the edge density in the Brownian co-graphon, Chapuy recently considered this problem in the case where $k=2$ and where the tree is the Brownian continuum random tree. We vastly extend this framework by considering general values of $k$ and general fragmentation trees, which include some prominent examples such as stable L\'evy trees and idealized models of phylogenetic trees. Other natural ancestral statistics are also considered. For a given tree model, we identify a phase transition-like phenomenon, with different asymptotic regimes for $N_k(n)$, depending on the position of $k$ relative to a model-dependent critical value.