A construction of a $\beta$-coalescent via the pruning of Binary Trees

TL;DR

This paper introduces a novel method to construct a Beta(3/2,1/2)-coalescent process using pruning procedures on both discrete binary trees and Aldous's continuum random tree, linking discrete and continuous models.

Contribution

It presents a new construction of the Beta-coalescent via pruning on binary trees and continuum trees, providing insights into its asymptotic behavior and coalescent events.

Findings

Constructed Beta-coalescent using pruning on binary trees

Extended the construction to continuum random trees

Derived asymptotic results on coalescent events

Abstract

Considering a random binary tree with $n$ labelled leaves, we use a pruning procedure on this tree in order to construct a $\beta(3/2,1/2)$-coalescent process. We also use the continuous analogue of this construction, i.e. a pruning procedure on Aldous's continuum random tree, to construct a continuous state space process that has the same structure as the $\beta$-coalescent process up to some time change. These two constructions unable us to obtain results on the coalescent process such as the asymptotics on the number of coalescent events or the law of the blocks involved in the last coalescent event.