Lambdacoalescents

Lambdacoalescents are a family of stochastic processes used to model genealogies in populations where several ancestral lineages can merge into a common ancestor in a single event. They generalize Kingman’s coalescent by allowing multiple mergers, rather than restricting to binary coalescent events. The family is defined by a finite measure Λ on the unit interval [0,1], which governs the rates of multiple-merger events.

For a sample of n labeled lineages, the process specifies the rate at which a merger involving k lineages (reducing the number of blocks from n to n−k+1) occurs. The rate at which any particular set of k lineages coalesces is ∫_0^1 x^{k-2} (1−x)^{n−k} dΛ(x). The total rate of such k-merger events is binom(n, k) times this integral, for k = 2, …, n. If Λ is a Dirac mass at 0 (the Kingman case), only binary mergers are possible, and the Lambda-coalescent reduces to Kingman’s coalescent.

Special cases among Lambdacoalescents include Beta-coalescents, where Λ is the Beta(2−α, α) distribution with 0 < α < 2. The parameter α controls the tendency for multiple mergers: smaller α values favor larger merger events, while α approaching 2 recovers behavior closer to Kingman’s coalescent. More general Λ measures give rise to a wide range of multiple-merger dynamics, including other Beta families and more intricate mixtures.

Lambdacoalescents are exchangeable coalescent processes on partitions of samples and produce ultrametric genealogies. Their long-term behavior, such as whether the process “comes down from infinity” or whether a persistent dust component remains, depends on the chosen Λ. They are used to model populations with highly skewed offspring distributions and to study allele frequency spectra and ancestral trees in such contexts.

See also: coalescent theory, Kingman coalescent, Beta-coalescent, Xi-coalescent.