Ancestral branching, cut-and-paste algorithms and associated tree and partition-valued processes

TL;DR

This paper introduces the ancestral branching algorithm for generating random fragmentation trees, along with a cut-and-paste procedure, providing a comprehensive framework for tree and partition-valued Markov processes with explicit transition probabilities.

Contribution

It presents a novel ancestral branching algorithm and a cut-and-paste transition procedure, linking tree and partition-valued Markov processes with explicit formulas and exchangeability properties.

Findings

Derived closed-form transition probabilities for the processes.

Established conditions for infinite exchangeability.

Analyzed equilibrium measures and associated processes.

Abstract

We introduce an algorithm for generating a random sequence of fragmentation trees, which we call the ancestral branching algorithm. This algorithm builds on the recursive partitioning structure of a tree and gives rise to an associated family of Markovian transition kernels whose finite-dimensional transition probabilities can be written in closed-form as the product over partition-valued Markov kernels. The associated tree-valued Markov process is infinitely exchangeable provided its associated partition-valued kernel is infinitely exchangeable. We also identify a transition procedure on partitions, called the cut-and-paste algorithm, which corresponds to a previously studied partition-valued Markov process on partitions with a bounded number of blocks. Specifically, we discuss the corresponding family of tree-valued Markov kernels generated by the combination of both the ancestral branching and cut-and-paste transition probabilities and show results for the equilibrium measure of this process, as well as its associated mass fragmentation-valued and weighted tree-valued processes.