Lambda- and Xi-coalescents
/ LAM-da and KSY (ksee) /
Kingman's coalescent merges lineages strictly two at a time, which is correct only when reproduction is mild. But some populations reproduce in jackpots — a single individual occasionally leaves a huge fraction of the next generation (think of a fish spawning millions of eggs, or a strongly favoured mutation sweeping a population). Then many present-day lineages can find their common ancestor in a single ancestral individual, all at once. Lambda-coalescents (and their generalization, Xi-coalescents) are the genealogical processes that allow such MULTIPLE and SIMULTANEOUS mergers, completing the classification of all exchangeable, consistent genealogies.
A Lambda-coalescent is a continuous-time Markov process on partitions in which, whenever there are b blocks, any specific subset of k of them (2 <= k <= b) merges into one at rate lambda_(b,k) = integral_0^1 x^(k-2) (1 - x)^(b-k) Lambda(dx), where Lambda is a fixed finite measure on [0, 1]. The measure Lambda is the single parameter that encodes the whole process (Pitman, Sagitov, Donnelly-Kurtz): an atom of Lambda at x means that, at the corresponding event, each lineage joins the merger independently with probability x — a fraction roughly x of all current lineages coalesce simultaneously into one block. Kingman's coalescent is the special case Lambda = delta_0 (a point mass at 0), which yields only pairwise mergers; the Bolthausen-Sznitman coalescent (central in spin glasses and the genealogy of branching random walk / BBM) is Lambda = Uniform[0,1]; and the Beta(2 - alpha, alpha)-coalescents arise as the genealogies of alpha-stable continuous-state branching processes (1 < alpha < 2), the dual on the genealogy side of the heavy-reproduction CSBPs. The Xi-coalescent (Mohle-Sagitov, Schweinsberg) is the further generalization permitting several DISTINCT groups of lineages to merge at the very same instant — needed when more than one parent simultaneously produces a macroscopic family — parametrized by a measure Xi on the infinite simplex.
Why it matters: Lambda- and Xi-coalescents are the right genealogical models for organisms with high fecundity and skewed offspring (many marine species, some viruses and pathogens) and for populations under strong, recurrent selection; their multiple-merger signature shows up in real DNA data as an excess of singleton mutations and a star-like genealogy that the Kingman model cannot fit. They are exactly the duals, on the backward-in-time genealogy side, of the continuous-state branching processes on the forward-in-time population side — the Lamperti and duality circle closes here. The honest content is that the choice between Kingman, Lambda, and Xi is a modelling decision dictated by the reproduction mechanism, not a free parameter: assume binary Kingman mergers for a high-fecundity population and you will systematically misdate common ancestors and misread selection. The measure Lambda (or Xi) is determined by how skewed the reproduction is, and inferring it from data is an active, subtle statistical problem.
In a marine species where one lucky spawn can dominate a year-class, dozens of sampled individuals may trace to a single recent ancestor in one event — a multiple merger. The genealogy is star-like rather than the slow pairwise tree of Kingman, and the Beta(2-alpha, alpha)-coalescent (dual to an alpha-stable CSBP) often fits the data, predicting an excess of rare (singleton) variants.
A measure Lambda on [0,1] encodes the merger rates; multiple (and, for Xi, simultaneous) mergers replace strict pairing.
Kingman is the special case Lambda = delta_0 (pairwise only); a general Lambda allows multiple simultaneous mergers, and Xi allows several distinct simultaneous mergers. Which one is correct is dictated by the reproduction mechanism, not chosen for convenience — using Kingman for high-fecundity species biases all inference.