Coalescents With Multiple Collisions
Annals of Probability
For each finite measure on 0 1 , a coalescent Markov process, with state space the compact set of all partitions of the set of positive integers, is constructed so the restriction of the partition to each finite subset of 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 1 0 x k−2 1 − x b−k dx . Call this process a -coalescent. Discrete measure-valued processes derived from the -coalescent
... l a system of masses undergoing coalescent collisions. Kingman's coalescent, which has numerous applications in population genetics, is the δ 0 -coalescent for δ 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 = U, and whenever an infinite number of masses are present, each collision in a -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 . The two-parameter Poisson-Dirichlet family of random discrete distributions derived from a stable subordinator, and corresponding exchangeable random partitions of 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.