Coalescents With Multiple Collisions

Abstract

For each finite measure $\Lambda$ on [0,1] a coalescent Markov process, with state space the compact set of all partitions of the set $\mathbb{N}$ of positive integers, is constructed so the restriction of the partition to each finite subset of $\mathbb{N}$ is a Markov chain with the following transition rates: when the partition has b blocks, each $k$-tuple of blocks is merging to form a single block at rate $\int_0^1x^{k-2}(1-x)^{b-k}\Lambda(dx)$. Call this process a $\Lambda$-coalescent. Discrete measure-valued processes derived from the $\Lambda$-coalescent model a system of masses undergoing coalescent collisions. Kingman’s coalescent, which has numerous applications in population genetics, is the $\delta_0$-coalescent for $\delta_0$ a unit mass at 0. The coalescent recently derived by Bolthausen and Sznitman from Ruelle’s probability cascades, in the context of the Sherrington–Kirkpatrick spin glass model in mathematical physics, is the $U$-coalescent for $U$ uniform on [0,1]. For $\Lambda=U$, and whenever an infinite number of masses are present, each collision in $\Lambda$-coalescent involves an infinite number of masses almost surely, and the proportion of masses involved exists as a limit almost surely and is distributed proportionally to $\Lambda$. The two-parameter Poisson–Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable random partitions of $\mathbb{N}$ governed by a generalization of the Ewens sampling formula, are applied to describe transition mechanisms for processes of coalescence and fragmentation, including the $U$-coalescent and its time reversal.