THE SPATIAL $\Lambda$-COALESCENT
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Vlada Limic and Anja Sturm
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This paper extends the notion of the $\Lambda$-coalescent of Pitman (1999)
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 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 $\Lambda$-coalescents on large tori in $d\ge 3$
dimensions.
Our results generalize and strengthen those of Greven et al.~(2005), who
studied
the spatial Kingman coalescent in this context.