Continuum asymptotics for tree growth models with uniform backward dynamics

TL;DR

This paper analyzes tree-valued Markov chains with uniform backward dynamics, showing they can be represented by weighted real trees and demonstrating their convergence to a random real tree under rescaling, extending previous binary case results.

Contribution

It introduces a full representation theorem for these Markov chains using weighted real trees with planar order and functions, and proves their convergence to a limiting real tree in the Gromov–Prokhorov metric.

Findings

Markov chains can be represented by weighted real trees.

Under rescaling, chains converge to a random real tree.

Generalizes previous binary case results.

Abstract

We study (plane) tree-valued Markov chains $(T_n,n \geq 1)$ with uniform backward dynamics and show that they can be obtained by sampling from a real tree. As non--plane trees, every such Markov chain is represented by a weighted real tree. We equip this real tree with a planar order as well as some extra functions for the full representation theorem. We also show that under an inhomogeneous rescaling after trimming leaves $(T_n, n\geq 1)$ converges to a random real tree in the Gromov--Prokhorov metric. This makes use of a special class of real trees, interval partition trees, which were introduced by Forman (2020). Moreover, this generalises and sheds some new light on work by Evans, Gr\"ubel and Wakolbinger (2017) on the binary special case.