Coalescent processes arising in a study of diffusive clustering

Abstract

This paper studies the spatial coalescent on $\Z^2$. In our setting, the partition elements are located at the sites of $\Z^2$ and undergo local delayed coalescence and migration. That is, pairs of partition elements located at the same site coalesce into one partition element after exponential waiting times. In addition, the partition elements perform independent random walks. The system starts in either locally finite configurations or in configurations containing countably many partition elements per site. These two situations are relevant if the coalescent is used to study the scaling limits for genealogies in Moran models respectively interacting Fisher-Wright diffusions (or Fleming-Viot processes), which is the key application of the present work. Our goal is to determine the longtime behavior with an initial population of countably many individuals per site restricted to a box $[-t^{α/2}, t^{α/2}]^2 \cap \Z^2$ and observed at time $t^β$ with $1 \geq β\geq α\ge 0$. We study both asymptotics, as $t\to\infty$, for a fixed value of $α$ as the parameter $β\in[α,1]$ varies, and for a fixed $β$, as the parameter $α\in [0,β]$ varies. This exhibits the genealogical structure of the mono-type clusters arising in 2-dimensional Moran and Fisher-Wright systems. (... for more see the actual preprint)