Coalescents with multiple collisions

For each finite measure Lambda on [0, 1], a coalescent Markov process, with state space the compact set of all partitions of the set N of positive integers, is constructed so the restriction of the partition to each finite subset of 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 integral(0)(1) x(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 a 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 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.