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.


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