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Coalescents with multiple collisions. (English) Zbl 0963.60079
This nice paper is concerned with a natural family of coalescents. These are Markov processes with values in the compact space of partitions of the set of positive integers, in which the random exchangeable partitions get coarser as time passes. The dynamics are described through the rate at which \(k\)-tuples of blocks are merging to form a single block. The author characterizes the family of admissible rates and provides a simple expression for these rates in terms of a certain finite measure \(\Lambda\) on \([0,1]\). Peharps the most important example of such processes is J. F. C. Kingman’s coalescent [Stochastic Processes Appl. 13, 235-248 (1982; Zbl 0491.60076)] which corresponds to the measure \(\Lambda=\delta_0\). Another remarkable example, corresponding to the case where \(\Lambda\) is the Lebesgue measure on \([0,1]\), is provided by the so-called \(U\)-coalescent [cf. E. Bolthausen and A.-S. Sznitman, Commun. Math. Phys. 197, No. 2, 247-276 (1998; Zbl 0927.60071)]. Several interesting properties of the \(U\)-coalescent are derived, in particular its transition mechanism is described in terms of the Poisson-Dirichlet distributions.

MSC:
60J75 Jump processes (MSC2010)
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