Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Connectedness of Poisson cylinders in Euclidean space

Erik I. Broman and Johan Tykesson

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We consider the Poisson cylinder model in $\mathbb{R}^{d}$, $d\ge3$. 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 (Probab. Theory Related Fields 154 (2012) 165–191). 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 $d-2$.


Nous considérons un modèle de cylindres suivant un processus de Poisson dans $\mathbb{R}^{d}$, $d\ge3$. Nous montrons que étant donnés deux cylindres ${\mathfrak{c}}_{1}$ et ${\mathfrak{c}}_{2}$ dans le processus, il y a une séquence d’au plus $d-2$ autres cylindres qui créent une connexion entre ${\mathfrak{c}}_{1}$ et ${\mathfrak{c}}_{2}$. En particulier, ceci montre que l’union des cylindres est un ensemble connecté, et répond à une question posée par (Probab. Theory Related Fields 154 (2012) 165–191). Nous montrons aussi qu’il y a des cylindres dans le processus qui ne sont pas connectés par une séquence d’au plus $d-3$ autres cylindres. Donc, le diamètre de l’amas de cylindres est égal à $d-2$.

Article information

Ann. Inst. H. Poincaré Probab. Statist., Volume 52, Number 1 (2016), 102-126.

Received: 17 September 2013
Revised: 9 May 2014
Accepted: 4 September 2014
First available in Project Euclid: 6 January 2016

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82B43: Percolation [See also 60K35]

Poisson cylinder model Continuum percolation


Broman, Erik I.; Tykesson, Johan. Connectedness of Poisson cylinders in Euclidean space. Ann. Inst. H. Poincaré Probab. Statist. 52 (2016), no. 1, 102--126. doi:10.1214/14-AIHP641.

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