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Percolation in the vacant set of Poisson cylinders. (English) Zbl 1263.82027
A Poisson point process is investigated on the space of random lines in \(\mathbb{R}^d\), \(d\geq 3\) where a multiplicative parameter \(u=u(d)\) labels low and high intensity regimes, including a percolation threshold (a critical value \(u_*\) of that parameter). Each line is interpreted as the axis of a bi-infinite cylinder of unit radius. Percolation is studied for the vacant set which is a random set \({\mathcal{V}} \subset \mathbb{R}^d\) not covered by the cylinders. The main question addressed is whether the set \({\mathcal{V}}\) has unbounded connected components , in which case we say that \({\mathcal{V}}\) percolates. Low and high intensity regimes are investigated. In particular, it is proved that in dimensions \(d\geq 3\), \({\mathcal{V}}\) does not percolate for high intensities. It is found that, for \(d\geq 4\), \({\mathcal{V}}\) does percolate at low intensities. Various inspirations follow from the work by A.-S. Sznitman [Ann. Math. (2) 171, No. 3, 2039–2087 (2010; Zbl 1202.60160)] on random interlacements, which share common features with Poisson cylinders. Specifically the infinite-range dependence of the current problem requires some techniques, previously developed by Sznitman.

MSC:
82B43 Percolation
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60G55 Point processes (e.g., Poisson, Cox, Hawkes processes)
82C41 Dynamics of random walks, random surfaces, lattice animals, etc. in time-dependent statistical mechanics
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