Abstract
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 $d\geq 3$ dimensions. Some of our results generalize and strengthen the corresponding results in Greven et al. (2005) concerning the spatial Kingman coalescent.
Highlights
The Λ-coalescent, sometimes called the coalescent with multiple collisions, is a Markov process Π whose state space is the set of partitions of the positive integers
The standard Λcoalescent Π starts at the partition of the positive integers into singletons, and its restriction to [n] := {1, . . . , n}, denoted by Πn, is the Λ-coalescent starting with n initial partition elements
The well-known Kingman coalescent [20] corresponds to the Λ-coalescent with Λ(dx) = δ0(dx), the unit atomic measure at 0
Summary
The Λ-coalescent, sometimes called the coalescent with multiple collisions, is a Markov process Π whose state space is the set of partitions of the positive integers. N} labeled by their location in G This will be stated precisely in Theorem 1 of Section 2 which is devoted to the construction of spatial Λ-coalescents Π with general (possibly infinite) initial states. In Lemma 8 and Proposition 11 we show that condition (4) implies P [#Π(t) < ∞, ∀t > 0] = 1, even if the initial configuration Π(0) contains infinitely many blocks In this case we say that the spatial Λ-coalescent comes down from infinity. Given Theorem 12, the approximation method of [16] section 7 applies verbatim and enables construction of the spatial Λ-coalescent on the whole lattice Zd so that even if started with infinitely many partition elements at each site of Zd, at all positive times the configuration is locally finite, almost surely. We obtain functional limit theorems for the partition structure and for the number of partitions, in Theorems 13 and 19 respectively
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