Scaling limits of multi-type Markov Branching trees

Abstract

We introduce multi-type Markov Branching trees, which are simple random population tree models where individuals are characterized by their size and their type and give rise to (size,type)-children in a Galton–Watson fashion, with the rule that the size of any individual is at least the sum of the sizes of its children. Assuming that the 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 case the types mix in the limit 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 built by gluing at each step on the current structure a finite tree picked randomly in a finite alphabet of trees, extending Rémy’s well-known algorithm for the generation of uniform binary trees to a fairly broad framework. We are then either in the critical regime with multi-type fragmentation trees in the scaling limit, or in the solo regime. 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 always multiple of the Brownian CRT, a pure monotype fragmentation tree in our framework.