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Abstract

Consider some population evolving stochastically in time. Conditional on the population surviving until some large time T, take a sample of individuals from those alive. What does the ancestral tree drawn out by this sample look like? Some special cases were known, e.g. Durrett (1978), O’Connell (1991), but we will discuss some more recent advances for Bienyame-Galton-Watson (BGW) branching processes conditioned to survive. In near-critical and in critical varying environment BGW settings, the same universal limiting sample genealogy always appears up to some deterministic time change (which only depends on the mean and variance of the offspring distributions). This genealogical tree has the same binary tree topology as the classical Kingman coalescent, but where the coalescent (or split) times are quite different due to stochastic population size effects, with a representation as a mixture of independent identically distributed times. In contrast, in critical infinite variance offspring settings, we find that more complex universal limiting sample genealogies emerge that exhibit multiple-mergers, these being driven by rare but massive birth events within the underlying population (eg. `superspreaders’ in an epidemic). Our key tool for proofs is a change of measure technique involving k distinguished particles, also known as spines. Some ongoing work, open problems, and potential downstream applications will also be mentioned.

This talk is based on some collaborative work with Juan Carlos Pardo (CIMAT), Samuel Johnston (Kings College London) in Annals of Probability (2024), with Sandra Palau (UNAM), J.C. Pardo in Annals of Applied Probability (2024), and with Matt Roberts (Bath), S. Johnston in Annals of Applied Probability (2020). I would also like to acknowledge the support of the New Zealand Royal Society Te Apārangi Marsden fund.