A tale of two balloons

TL;DR

This paper investigates a stochastic growth and popping process of balloons originating from Poisson points in different geometries, establishing conditions for infinite coverage and introducing a new 0-1 law for stationary processes.

Contribution

It provides the first analysis of balloon growth and popping in Euclidean, hyperbolic, and tree spaces, including a novel 0-1 law and bounds on well-separated sets.

Findings

In Euclidean space, the process results in either infinite or finite coverage depending on parameters.

A new 0-1 law for stationary processes is established.

Upper bounds on densities of well-separated sets in regular trees are proved.

Abstract

From each point of a Poisson point process start growing a balloon at rate 1. When two balloons touch, they pop and disappear. Is every point contained in balloons infinitely often or not? We answer this for the Euclidean space, the hyperbolic plane and regular trees. The result for the Euclidean space relies on a novel 0-1 law for stationary processes. Towards establishing the results for the hyperbolic plane and regular trees, we prove an upper bound on the density of any well-separated set in a regular tree which is a factor of an i.i.d. process.