Critical cluster cascades

TL;DR

This paper studies a sequence of Poisson cluster processes where cluster sizes grow via branching random walks, showing conditions under which the process either becomes extinct or persists with a non-zero intensity.

Contribution

It establishes the weak convergence of critical cluster cascades and characterizes conditions for their persistence or extinction, providing new insights into their long-term behavior.

Findings

The process converges weakly to either the void process or a process with the same intensity.

Persistence occurs if the Palm version of the outgrown branching random walk is locally finite.

The paper provides numerous examples of persistent critical cluster cascades.

Abstract

We consider a sequence of Poisson cluster point processes on $\mathbb{R}^d$: at step $n\in\mathbb{N}_0$ of the construction, the cluster centers have intensity $c/(n+1)$ for some $c>0$, and each cluster consists of the particles of a branching random walk up to generation $n$ generated by a point process with mean 1. We show that this 'critical cluster cascade' converges weakly, and that either the limit point process equals the void process (extinction), or it has the same intensity $c$ as the critical cluster cascade (persistence). We obtain persistence, if and only if the Palm version of the outgrown critical branching random walk is locally a.s. finite. This result allows us to give numerous examples for persistent critical cluster cascades.