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Cylinders’ percolation in three dimensions. (English) Zbl 1333.60205
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 \(u\).

60K35 Interacting random processes; statistical mechanics type models; percolation theory
60K37 Processes in random environments
Full Text: DOI
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