Restricted exchangeable partitions and embedding of associated hierarchies in continuum random trees

TL;DR

This paper introduces restricted exchangeable partitions, explores their properties, and demonstrates their applications in embedding hierarchies into continuum random trees, including convergence results for specific models like Ford's alpha model.

Contribution

It develops the theory of restricted exchangeable partitions and applies it to embedding hierarchies into continuum random trees, including proving a conjectured limit for Ford's alpha model.

Findings

Established integral representations for restricted exchangeable partitions

Demonstrated convergence of associated hierarchies to continuum random trees

Proved a limit result for Ford's alpha model and its extension

Abstract

We introduce the notion of a restricted exchangeable partition of $\mathbb{N}$. We obtain integral representations, consider associated fragmentations, embeddings into continuum random trees and convergence to such limit trees. In particular, we deduce from the general theory developed here a limit result conjectured previously for Ford's alpha model and its extension, the alpha-gamma model, where restricted exchangeability arises naturally.