Cylinders' percolation in three dimensions

Marcelo Hilário, Vladas Sidoravicius, Augusto Teixeira

Published 2012-02-08Version 1

We study the complementary set of a Poissonian ensemble of infinite cylinders in 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 R^3, regardless of the choice of u.