## The spatial $$\Lambda$$-coalescent.(English)Zbl 1113.60077

Summary: This paper extends the notion of the $$\Lambda$$-coalescent of J. Pitman [Ann. Probab. 27, No. 4, 1870–1902 (1999; Zbl 0963.60079)] to the spatial setting. The partition elements of the spatial $$\Lambda$$-coalescent migrate in a (finite) geographical space and may only coalesce if located at the same site of the space. We characterize the $$\Lambda$$-coalescents that come down from infinity, in an analogous way to J. Schweinsberg [Electron. Commun. Probab. 5, 1–11 (2000; Zbl 0953.60072)]. Surprisingly, all spatial coalescents that come down from infinity, also come down from infinity in a uniform way. This enables us to study space-time asymptotics of spatial $$\Lambda$$-coalescents on large tori in $$d\geq 3$$ dimensions. Some of our results generalize and strengthen the corresponding results of A. Greven, V. Limic and A. Winter [Electron J. Probab. 10, Paper No. 39, 1286–1358 (2005; Zbl 1109.60082)] concerning the spatial Kingman coalescent.

### MSC:

 60J25 Continuous-time Markov processes on general state spaces 60K35 Interacting random processes; statistical mechanics type models; percolation theory

### Citations:

Zbl 0963.60079; Zbl 0953.60072; Zbl 1109.60082
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