Abstract
We consider a Poisson point process on the space of lines in \({{\mathbb R}^d}\), where a multiplicative factor u > 0 of the intensity measure determines the density of lines. Each line in the process is taken as the axis of a bi-infinite cylinder of radius 1. We investigate percolative properties of the vacant set, defined as the subset of \({{\mathbb R}^d}\) that is not covered by any such cylinder. We show that in dimensions d ≥ 4, there is a critical value \({u_*(d) \in (0,\infty)}\), such that with probability 1, the vacant set has an unbounded component if u < u *(d), and only bounded components if u > u *(d). For d = 3, we prove that the vacant set does not percolate for large u and that the vacant set intersected with a two-dimensional subspace of \({{\mathbb R}^d}\) does not even percolate for small u > 0.
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J. Tykesson’s research was supported by a post-doctoral grant of the Swedish Research Council.
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Tykesson, J., Windisch, D. Percolation in the vacant set of Poisson cylinders. Probab. Theory Relat. Fields 154, 165–191 (2012). https://doi.org/10.1007/s00440-011-0366-3
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DOI: https://doi.org/10.1007/s00440-011-0366-3
Mathematics Subject Classification (2000)
- 60K35
- 82B43