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Peer-reviewedCC BY-NC 4.0The Annals of Probability

Almost sharp sharpness for Poisson Boolean percolation

  • Barbara Dembin1
  • Vincent Tassion2
  1. 1IRMA, CNRS and Université de Strasbourg
  2. 2D-MATH, ETH Zürich
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Abstract

We consider Poisson Boolean percolation on Rd with power-law distribution on the radius with a finite d-moment for d≥2. We prove that subcritical sharpness occurs for all but a countable number of power-law distributions. This extends the results of Duminil-Copin–Raoufi–Tassion (Ann. Henri Lebesgue 3 (2020) 677–700) where subcritical sharpness is proved under the assumption that the radii distribution has a 5d−3 finite moment. Our proofs techniques are different from (Ann. Henri Lebesgue 3 (2020) 677–700): we do not use a randomized algorithm and rely on specific independence properties of Boolean percolation, inherited from the underlying Poisson process. We also prove supercritical sharpness for any distribution with a finite d-moment and the continuity of the critical parameter for the truncated distribution when the truncation goes to infinity.

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Cite

BibTeXRIS
BibTeX
@article{dembin2026almost,
  title = {{Almost sharp sharpness for Poisson Boolean percolation}},
  author = {Barbara Dembin and Vincent Tassion},
  journal = {The Annals of Probability},
  year = {2026},
  month = jun,
  doi = {10.1214/25-aop1771},
  url = {https://hal.science/hal-05826940v1},
  note = {CC BY-NC 4.0},
}
RIS
TY  - JOUR
TI  - Almost sharp sharpness for Poisson Boolean percolation
AU  - Barbara Dembin
AU  - Vincent Tassion
PY  - 2026
DA  - 2026/06/30
JO  - The Annals of Probability
DO  - 10.1214/25-aop1771
UR  - https://hal.science/hal-05826940v1
AB  - We consider Poisson Boolean percolation on Rd with power-law distribution on the radius with a finite d-moment for d≥2. We prove that subcritical sharpness occurs for all but a countable number of power-law distributions. This extends the results of Duminil-Copin–Raoufi–Tassion (Ann. Henri Lebesgue 3 (2020) 677–700) where subcritical sharpness is proved under the assumption that the radii distribution has a 5d−3 finite moment. Our proofs techniques are different from (Ann. Henri Lebesgue 3 (2020) 677–700): we do not use a randomized algorithm and rely on specific independence properties of Boolean percolation, inherited from the underlying Poisson process. We also prove supercritical sharpness for any distribution with a finite d-moment and the continuity of the critical parameter for the truncated distribution when the truncation goes to infinity.
SN  - 0091-1798
SN  - 2168-894X
ER  -