A tale of two balloons

Omer Angel, Gourab Ray*, Yinon Spinka

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review


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.

Original languageEnglish
Pages (from-to)815-837
Number of pages23
JournalProbability Theory and Related Fields
Issue number3-4
StatePublished - Apr 2023


FundersFunder number
Natural Sciences and Engineering Research Council of Canada50311-57400


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