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The spatial -coalescent

This paper extends the notion of the Λ-coalescent of Pitman (1999) to the spatial setting. The partition elements of the spatial Λ-coalescent migrate in a (ﬁnite) geographical space and may only coalesce if located at the same site of the space. We characterize the Λ-coalescents that come down from inﬁnity, in an analogous way to Schweinsberg (2000). 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 Λ-coalescents on large tori in d ≥ 3 dimensions. Some of our results generalize and strengthen the corresponding results in Greven et al. (2005) concerning the spatial Kingman coalescent.