an:06511022
Zbl 1333.60205
Hil??rio, M. R.; Sidoravicius, V.; Teixeira, A.
Cylinders' percolation in three dimensions
EN
Probab. Theory Relat. Fields 163, No. 3-4, 613-642 (2015).
00349790
2015
j
60K35 60K37
percolation; infinite cylinders; phase transition
Summary: We study the complementary set of a Poissonian ensemble of infinite cylinders in \({\mathbb {R}}^3\), for which an intensity parameter \(u > 0\) controls the amount of cylinders to be removed from the ambient space. We establish a non-trivial phase transition, for the existence of an unbounded connected component of this set, as \(u\) crosses a critical non-degenerate intensity \(u_*\). We moreover show that this complementary set percolates in a sufficiently thick slab, in spite of the fact that it does not percolate in any given plane of \({\mathbb {R}}^3\), regardless of the choice of \(u\).