Xicoalescents

Xicoalescents, or Xi-coalescents, are a class of stochastic processes used in population genetics to model the genealogical history of a sample when multiple ancestral lineages may merge in a single coalescent event. They generalize the classical Kingman coalescent by allowing simultaneous merging of more than two ancestral lines, a feature that captures skewed reproductive success observed in some populations. The rate and sizes of these events are governed by a finite measure Ξ on the infinite simplex, which encodes how many lineages can merge at once and with what probabilities.

In a Xi-coalescent, the evolution is described by a Poissonian construction: at random times defined by a Poisson point process, blocks of current ancestral lineages merge according to a pattern drawn from Ξ. The model thus allows simultaneous mergers of multiple blocks, not just pairwise coalescences. In appropriate limiting choices of Ξ, Xi-coalescents reduce to Lambda-coalescents, and in particular to the Kingman coalescent when the measure concentrates on binary mergers.

History and development: The Xi-coalescent framework was developed in the late 1990s as part of broader work on coalescent processes with multiple collisions, with key contributions by Pitman, Sagitov, and Schweinsberg. Since then, the Xi-coalescent has become a standard model in the study of genealogies under skewed offspring distributions and has been extended to include various demographic and recombination settings.

Applications and interpretation: Xi-coalescents are used to model populations with sweepstakes reproduction, such as certain marine species or viral populations under strong selection, where many lineages can coalesce at once. They provide a theoretical bridge between Kingman and more general multiple-merger models and inform statistical methods for inferring population history under non-Poissonian reproduction patterns.