an:06550132
Zbl 1333.60197
Broman, Erik I.; Tykesson, Johan
Connectedness of Poisson cylinders in Euclidean space
EN
Ann. Inst. Henri Poincar??, Probab. Stat. 52, No. 1, 102-126 (2016).
00352734
2016
j
60K35 60D05 82B43
Poisson cylinder model; continuum percolation
Summary: We consider the Poisson cylinder model in \(\mathbb{R}^{d}\), \(d\geq3\). We show that given any two cylinders \({\mathfrak{c}}_{1}\) and \({\mathfrak{c}}_{2}\) in the process, there is a sequence of at most \(d-2\) other cylinders creating a connection between \({\mathfrak{c}}_{1}\) and \({\mathfrak{c}}_{2}\). In particular, this shows that the union of the cylinders is a connected set, answering a question appearing in [\textit{J. Tykesson} and \textit{D. Windisch},Probab. Theory Relat. Fields 154, No. 1--2, 165--191 (2012; Zbl 1263.82027)]. We also show that there are cylinders in the process that are not connected by a sequence of at most \(d-3\) other cylinders. Thus, the diameter of the cluster of cylinders equals \(d-2\).
Zbl 1263.82027