Coming down from infinity for coordinated particle systems

TL;DR

This paper investigates the conditions under which coordinated particle systems, where particles can coalesce, migrate, reproduce, and die simultaneously, transition from infinitely many particles to finitely many in finite time.

Contribution

It provides necessary and sufficient conditions for coming down from infinity in coordinated particle systems, extending understanding beyond classical coalescent models.

Findings

Identifies conditions for coming down from infinity in these systems

Generalizes known results from $ ext{Lambda}$-coalescents

Offers a framework for analyzing particle systems with coordinated events

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.