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.
This is a preview of subscription content, access via your institution.


References
Aldous, D., Lyons, R.: Processes on unimodular random networks. Electron. J. Probab. 12, 1454–1508 (2007)
Bollobás, B.: The independence ratio of regular graphs. In: Proceedings of the American Mathematical Society, pp. 433–436 (1981)
Häggström, O., Meester, R.: Nearest neighbor and hard sphere models in continuum percolation. Random Struct. Algor. 9(3), 295–315 (1996)
Holroyd, A.E., Pemantle, R., Peres, Y., Schramm, O.: Poisson matching. In: Annales de l’IHP Probabilités et statistiques, volume 45, pp. 266–287 (2009)
Lauer, J., Wormald, N.: Large independent sets in regular graphs of large girth. J. Combin. Theory Ser. B 97(6), 999–1009 (2007)
McKay, B.D.: Independent sets in regular graphs of high girth. Ars Combin. 23, 179–185 (1987)
Rahman, M., Virag, B.: Local algorithms for independent sets are half-optimal. Ann. Probab. 45(3), 1543–1577 (2017)
Roth, M.: A Tale of Five Balloons. Poalim Library (1974)
Tanny, D.: On branching processes in random environments and other related growth processes. PhD thesis, Cornell University (1974)
Tanny, D.: A zero-one law for stationary sequences. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 30(2), 139–148 (1974)
Vitali, G.: Sui gruppi di punti e sulle funzioni di variabili reali. Atti Accad. Sci. Torino 43, 75–92 (1908)
Acknowledgements
We would like to thank Itai Benjamini for suggesting this problem. We are grateful to Thomas Budzinski for an elegant proof of Theorem 4, and to Ofer Zeitouni for bringing [10] to our attention. We would also like to thank the referee for useful comments. The research of GR is supported by NSERC 50311-57400.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Angel, O., Ray, G. & Spinka, Y. A tale of two balloons. Probab. Theory Relat. Fields 185, 815–837 (2023). https://doi.org/10.1007/s00440-022-01165-6
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-022-01165-6
Mathematics Subject Classification
- 60G55
- 60G10