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A tale of two balloons

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.

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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.

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Correspondence to Gourab Ray.

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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

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  • DOI: https://doi.org/10.1007/s00440-022-01165-6

Mathematics Subject Classification

  • 60G55
  • 60G10