Coming down from infinity for coordinated particle systems

Abstract

We study coming down from infinity for coordinated particle systems. In a coordinated particle system, particles live on a set of sites $V$ and are able to coalesce, migrate, reproduce, and die. The dynamics of these events are coordinated in that many particles may undergo the same action simultaneously. Coming down from infinity is the phenomenon where a process starting with infinitely many particles will almost surely have only finitely many particles after any positive time. This phenomenon can be observed in some $\Lambda$-coalescents, and we give necessary and sufficient conditions to observe coming down from infinity in coordinated particle systems.