## Electronic Journal of Probability

### The spatial $\Lambda$-coalescent

#### 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.

#### Article information

Source
Electron. J. Probab., Volume 11 (2006), paper no. 15, 363-393.

Dates
Accepted: 19 May 2006
First available in Project Euclid: 31 May 2016

https://projecteuclid.org/euclid.ejp/1464730550

Digital Object Identifier
doi:10.1214/EJP.v11-319

Mathematical Reviews number (MathSciNet)
MR2223040

Zentralblatt MATH identifier
1113.60077

Rights
Limic, Vlada; Sturm, Anja. The spatial $\Lambda$-coalescent. Electron. J. Probab. 11 (2006), paper no. 15, 363--393. doi:10.1214/EJP.v11-319. https://projecteuclid.org/euclid.ejp/1464730550