multiplemerger

Multiple merger coalescents are a class of stochastic processes used in population genetics to describe the ancestral relationships among a sample of genes when more than two lineages can merge at a single time. Unlike Kingman’s coalescent, which permits only binary mergers, multiple merger models allow events where several lineages trace back to a common ancestor in one coalescent event. This leads to genealogies that can be star-like or have long, simultaneous coalescence patterns.

The most common formal framework is the Lambda-coalescent. In this setting, the rates of merging are governed by a finite measure Lambda on the interval [0,1]. At a merge event, a random fraction x of the current lineages may coalesce into one ancestor, with the distribution of such events determined by Lambda. A related generalization is the Xi-coalescent, which allows multiple simultaneous mergers of several groups of lineages.

Multiple merger models arise as scaling limits in populations with highly skewed offspring distributions or abrupt selective events. They are used to model situations where a single individual can contribute a large fraction of the next generation, such as sweepstakes reproduction, rapid adaptation, certain viral populations, or cancer cell lineages. These processes produce genealogies with different shapes and expected site frequency spectra compared to Kingman’s coalescent.

Canonical examples include the Bolthausen–Sznitman coalescent, a Lambda-coalescent with Lambda uniformly distributed on [0,1], which captures strong selection scenarios. Kingman’s coalescent itself is recovered as a special case when only binary mergers are allowed.

See also coalescent theory, Lambda-coalescent, Xi-coalescent, Bolthausen–Sznitman coalescent.