# Gibbs fragmentation trees

## TL;DR

This paper characterizes Gibbs fragmentation trees, linking binary cases to Aldous' beta-splitting model and multifurcating cases to Poisson-Dirichlet models, expanding parameter ranges for both.

## Contribution

It identifies the most general Gibbs-type fragmentation trees in binary and multifurcating cases, extending existing models with broader parameter ranges.

## Key findings

- Binary Gibbs fragmentation trees match Aldous' beta-splitting model with extended parameters.
- Multifurcating Gibbs fragmentation trees relate to two-parameter Poisson-Dirichlet models.
- Extended parameter ranges provide a more comprehensive understanding of fragmentation processes.

## Abstract

We study fragmentation trees of Gibbs type. In the binary case, we identify the most general Gibbs-type fragmentation tree with Aldous' beta-splitting model, which has an extended parameter range $\beta>-2$ with respect to the ${\rm beta}(\beta+1,\beta+1)$ probability distributions on which it is based. In the multifurcating case, we show that Gibbs fragmentation trees are associated with the two-parameter Poisson--Dirichlet models for exchangeable random partitions of $\mathbb {N}$, with an extended parameter range $0\le\alpha\le1$, $\theta\ge-2\alpha$ and $\alpha<0$, $\theta =-m\alpha$, $m\in \mathbb {N}$.