Scaling limits of multi-type Markov Branching trees

TL;DR

This paper introduces multi-type Markov Branching trees and characterizes their scaling limits in different regimes, unifying various models and providing new insights into the behavior of large multi-type trees.

Contribution

It develops a unified framework for the scaling limits of multi-type Markov Branching trees, including new regimes and applications to random tree models and Galton-Watson trees.

Findings

In the critical regime, multi-type fragmentation trees emerge as limits.

In the solo regime, limits are monotype trees with no type change.

In the mixing regime, limits are scaled Brownian CRTs.

Abstract

We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and type and give rise to (size,type)-children in a Galton-Watson fashion, with the rule that the size of any individual is a least the sum of the sizes of its children. Assuming that macroscopic size-splittings are rare, we describe the scaling limits of multi-type Markov Branching trees in terms of multi-type self-similar fragmentation trees. We observe three different regimes according to whether the probability of type change of a size-biased child is proportional to the probability of macroscopic splitting (the critical regime, in which we get in the limit multi-type fragmentation trees with indeed several types), smaller than the probability of macroscopic splitting (the solo regime, in which the limit trees are monotype as we never see a type change), or larger than the probability of macroscopic splitting (the mixing regime, in which the types mix and we get monotype fragmentation trees). This framework allows us to unify models which may a priori seem quite different, a strength which we illustrate with two notable applications. The first one concerns the description of the scaling limits of growing models of random trees that extend R\'emy's well-known algorithm for the generation of uniform binary trees to a fairly broad framework. We are then either in the critical or in the solo regimes. The second application concerns the scaling limits of large multi-type critical Galton-Watson trees when the offspring distributions all have finite second moments. This topic has already been studied but our approach gives a different proof and we improve on previous results by relaxing some hypotheses. We are then in the mixing regime: the scaling limits are multiple of the Brownian CRT, a pure monotype fragmentation tree in our framework.