The genealogy of Galton-Watson trees

TL;DR

This paper investigates the detailed genealogical structure of a continuous-time Galton-Watson tree when sampling multiple particles at a fixed time, extending previous work on pairwise relationships to the general case.

Contribution

It provides a comprehensive analysis of the genealogical trees for multiple particles in continuous-time Galton-Watson processes, generalizing prior results focused on pairs.

Findings

Characterization of the full genealogical structure for arbitrary k

Extension of near-critical asymptotics to general k

Unified framework for understanding particle genealogies

Abstract

Take a continuous-time Galton-Watson tree and pick $k$ distinct particles uniformly from those alive at a time $T$. What does their genealogical tree look like? The case $k=2$ has been studied by several authors, and the near-critical asymptotics for general $k$ appear in Harris, Johnston and Roberts (2018). Here we give the full picture.