Abstract
We prove phase transitions for continuum percolation in a Boolean model based on a Cox point process with nonstabilizing directing measure. The directing measure, which can be seen as a stationary random environment for the classical Poisson–Boolean model, is given by a planar rectangular Poisson line process. This Manhattan grid type construction features long-range dependencies in the environment, leading to absence of a sharp phase transition for the associated Cox–Boolean model. The phase transitions are established under individually as well as jointly varying parameters. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [Hof05].
Funding Statement
This work was supported by the German Research Foundation under Germany’s Excellence Strategy MATH+: The Berlin Mathematics Research Center, EXC-2046/1 project ID: 390685689, and the Leibniz Association within the Leibniz Junior Research Group on Probabilistic Methods for Dynamic Communication Networks as part of the Leibniz Competition.
Acknowledgments
The authors like to thank Alexandre Stauffer for fruitful discussions.
Citation
Benedikt Jahnel. Sanjoy Kumar Jhawar. Anh Duc Vu. "Continuum percolation in a nonstabilizing environment." Electron. J. Probab. 28 1 - 38, 2023. https://doi.org/10.1214/23-EJP1029
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