Multi-type fragmentation trees as scaling limits of Markov branching trees
随机数学中心学术报告
报告题目(Title)：Multi-type fragmentation trees as scaling limits of Markov branching trees
报告人(Speaker)：Robin Stephenson (University of Oxford)
地点(Place)：后主楼12层1220B
时间(Time)：2019年12月17日 10:00-11:00
邀请人(Inviter)：何辉
报告摘要
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 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 fragmentation trees and observe two main different regimes depending on how the rate of type change and the rate of macroscopic splits in a typical path compare. 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, and the second 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.
