Local limits of Markov Branching trees and their volume growth

TL;DR

This paper investigates the local limits of Markov branching trees, establishing convergence to infinite trees and self-similar fragmentation trees, and analyzing volume growth, with applications to models like Galton-Watson trees.

Contribution

It provides a unified framework for understanding the local limits and volume growth of Markov branching trees, including new convergence results and applications to various tree models.

Findings

Convergence of Markov branching trees to infinite trees under certain conditions

Conditions for convergence to self-similar fragmentation trees with immigration

Asymptotic volume growth of the trees' balls centered at the root

Abstract

We are interested in the local limits of families of random trees that satisfy the Markov branching property, which is fulfilled by a wide range of models. Loosely, this property entails that given the sizes of the sub-trees above the root, these sub-trees are independent and their distributions only depend upon their respective sizes. The laws of the elements of a Markov branching family are characterised by a sequence of probability distributions on the sets of integer partitions which describes how the sizes of the sub-trees above the root are distributed. We prove that under some natural assumption on this sequence of probabilities, when their sizes go to infinity, the trees converge in distribution to an infinite tree which also satisfies the Markov branching property. Furthermore, when this infinite tree has a single path from the root to infinity, we give conditions to ensure its convergence in distribution under appropriate rescaling of its distance and counting measure to a self-similar fragmentation tree with immigration. In particular, this allows us to determine how, in this infinite tree, the "volume" of the ball of radius $R$ centred at the root asymptotically grows with $R$. Our unified approach will allow us to develop various new applications, in particular to different models of growing trees and cut-trees, and to recover known results. An illustrative example lies in the study of Galton-Watson trees: the distribution of a critical Galton-Watson tree conditioned on its size converges to that of Kesten's tree when the size grows to infinity. If furthermore, the offspring distribution has finite variance, under adequate rescaling, Kesten's tree converges to Aldous' self-similar CRT and the total size of the $R$ first generations asymptotically behaves like $R^2$.