The coalescent structure of Galton–Watson trees in varying environments

Abstract

We investigate the genealogy of a sample of $\mathit{k}\ge 2$ particles chosen uniformly without replacement from a population alive at large times in a critical discrete-time Galton–Watson process in a varying environment (GWVE). We will show that subject to an explicit deterministic time-change involving only the mean and variances of the varying offspring distributions, the sample genealogy always converges to the same universal genealogical structure; it has the same tree topology as Kingman’s coalescent, and the coalescent times of the $\mathit{k}-1$ pairwise mergers look like a mixture of independent identically distributed times. Our approach uses k distinguished spine particles and a suitable change of measure under which (a) the spines form a uniform sample without replacement, as required, but additionally (b) there is k-size biasing and discounting according to the population size. Our work significantly extends the spine techniques developed in Harris, Johnston and Roberts (Ann. Appl. Probab. (2020) 30 1368–1414) for genealogies of uniform samples of size k in near-critical continuous-time Galton–Watson processes, as well as a two-spine GWVE construction in Cardona and Palau (Bernoulli (2021) 27 1643–1665). Our results complement recent works by Kersting (Proc. Steklov Inst. Maths. (2022) 316 209–219) and Boenkost, Foutel-Rodier and Schertzer (arXiv: 2207.11612).