Percolation in the vacant set of Poisson cylinders

Details Percolation in the vacant set of Poisson cylinders Percolation in the vacant set of Poisson cylinders

2010-10-26

arxiv.org/abs/1010.5338v2

Probability Theory and Related Fields, 2011

Mathematics

Probability

29 pages, 3 figures

Scientific paper

10.1007/s00440-011-0366-3

We consider a Poisson point process on the space of lines in 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 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 uu_*(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 R^d does not even percolate for small u>0.

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