Abstract
We consider the Boolean model with random radii based on Cox point processes. Under a condition of stabilization for the random environment, we establish existence and non-existence of subcritical regimes for the size of the cluster at the origin in terms of volume, diameter and number of points. Further, we prove uniqueness of the infinite cluster for sufficiently connected environments.
Acknowledgements
The authors thank A. Hinsen, C. Hirsch and W. König for interesting discussions and comments. The authors also thank three anonymous reviewers for their very useful comments. In particular, we would like to thank one anonymous reviewer for suggesting (i) the reference Gouéré (2009), which provides an alternative proof strategy for the Part (2) and the first part of Part (3) of Theorem 8, and (ii) an alternative approach to the proof of the second part of Part (3) of Theorem 8 via a simplified version of Condition (2.1).
This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy MATH+: The Berlin Mathematics Research Center, EXC-2046/1 project ID: 390685689 and by Orange S.A.
Citation
Benedikt Jahnel. András Tóbiás. Elie Cali. "Phase transitions for the Boolean model of continuum percolation for Cox point processes." Braz. J. Probab. Stat. 36 (1) 20 - 44, March 2022. https://doi.org/10.1214/21-BJPS514
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