A phase transition for the heights of a fragmentation tree

TL;DR

This paper investigates the asymptotic behavior of fragmentation trees, revealing a phase transition in the shattering times, with implications for understanding the dynamics of random split trees.

Contribution

It establishes a phase transition in the asymptotic regimes of shattering times for homogeneous fragmentation processes, extending understanding of their long-term behavior.

Findings

Identifies a phase transition in the asymptotic behavior of shattering times.

Provides a framework applicable to the analysis of random split trees.

Characterizes the regimes of fragmentation dynamics over time.

Abstract

We provide information about the asymptotic regimes for a homogeneous fragmentation of a finite set. We establish a phase transition for the asymptotic behaviours of the shattering times, defined as the first instants when all the blocks of the partition process have cardinality less than a fixed integer. Our results may be applied to the study of certain random split trees.