Universality classes for the coalescent structure of heavy-tailed Galton-Watson trees

Abstract

Consider a population evolving as a critical continuous-time Galton-Watson (GW) tree. Conditional on the population surviving until a large time $T$, sample $k$ individuals uniformly at random (without replacement) from amongst those alive at time $T$. What is the genealogy of this sample of individuals? In cases where the offspring distribution has finite variance, the probabilistic properties of the joint ancestry of these $k$ particles are well understood, as seen in \cite{HJR20, J19}. In the present article, we study the joint ancestry of a sample of $k$ particles under the following regime: the offspring distribution has mean $1$ (critical) and the tails of the offspring distribution are \emph{heavy} in that $\alpha \in (1,2]$ is the supremum over indices $\beta$ such that the $\beta^{\text{th}}$ moment is finite. We show that for each $\alpha$, after rescaling time by $1/T$, there is a universal stochastic process describing the joint coalescent structure of the $k$ distinct particles. The special case $\alpha = 2$ generalises the known case of sampling from critical GW trees with finite variance where only pairwise mergers are observed and the genealogical tree is, roughly speaking, some kind of mixture of time-changed Kingman coalescents. The cases $\alpha \in (1,2)$ introduce new universal limiting partition-valued stochastic processes with interesting probabilistic structures which have representations connected to the Lauricella function and the Dirichlet distribution, and whose coalescent structures exhibit multiple-mergers of family lines. Moreover, in the case $\alpha \in (1,2)$, we show that the coalescent events of the ancestry of the $k$ particles are associated with birth events that produce giant numbers of offspring of the same order of magnitude as the entire population size, and we compute the joint law of the ancestry together with the sizes of these giant births.