Continuum percolation in a nonstabilizing environment
- Jahnel, Benedikt
- Jhawar, Sanjoy Kumar
- Vu, Anh Duc
2020 Mathematics Subject Classification
- 60K35 60K37
- Boolean model, Cox point process, Manhattan grid, discretization, phase transition
We prove nontrivial 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. Our proofs rest on discretization arguments and a comparison to percolation on randomly stretched lattices established in [MR2116736].