Rate of coalescence of pairs of lineages in the spatial {\lambda}-Fleming-Viot process

TL;DR

This paper analyzes the coalescence time of two lineages in the spatial λ-Fleming-Viot process, providing approximation schemes and addressing the 'coming down from infinity' property in different spatial settings.

Contribution

It introduces new approximation methods for coalescence times and investigates the genealogical process behavior in spatial models.

Findings

Derived differential equations linking coalescence time and spatial distance.

Provided approximation schemes for coalescence times in different spatial domains.

Addressed the 'coming down from infinity' property in the spatial λ-Fleming-Viot process.

Abstract

We revisit the spatial ${\lambda}$-Fleming-Viot process introduced in [1]. Particularly, we are interested in the time $T_0$ to the most recent common ancestor for two lineages. We distinguish between the case where the process acts on the entire two-dimensional plane, and on a finite rectangle. Utilizing a differential equation linking $T_0$ with the physical distance between the lineages, we arrive at simple and reasonably accurate approximation schemes for both cases. Furthermore, our analysis enables us to address the question of whether the genealogical process of the model "comes down from infinity", which has been partly answered before in [2].