Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

Connectedness of Poisson cylinders in Euclidean space

Erik I. Broman and Johan Tykesson

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

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 $d-2$.

Résumé

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

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

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

Permanent link to this document
https://projecteuclid.org/euclid.aihp/1452089262

Digital Object Identifier
doi:10.1214/14-AIHP641

Mathematical Reviews number (MathSciNet)
MR3449296

Zentralblatt MATH identifier
1179.62076

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

Keywords
Poisson cylinder model Continuum percolation

Citation

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. https://projecteuclid.org/euclid.aihp/1452089262


Export citation

References

  • [1] I. Benjamini. Personal communication.
  • [2] I. Benjamini, H. Kesten, Y. Peres and O. Schramm. Geometry of the uniform spanning forest: Transitions in dimensions 4, 8, 12, …. Ann. of Math. (2) 160 (2) (2004) 465–491.
  • [3] G. Grimmett. Percolation, 2nd edition. Springer, Berlin, 1999.
  • [4] T. Hara, R. van der Hofstad and G. Slade. Critical two-point functions and the lace expansion spread-out high-dimensional percolation and related models. Ann. Probab. 31 (2003) 349–408.
  • [5] M. Hilário, V. Sidoravicius and A. Teixeira. Cylinders’ percolation in three dimensions. Probab. Theory Related Fields 163 (2015) 613–642.
  • [6] H. Lacoin and J. Tykesson. On the easiest way to connect $k$ points in the random interlacements process. ALEA Lat. Am. J. Probab. Math. Stat. 10 (2013) 505–524.
  • [7] R. Meester and R. Roy. Continuum Percolation. Cambridge Univ. Press, Cambridge, 1996.
  • [8] E. Procaccia and J. Tykesson. Geometry of the random interlacement. Electron. Commun. Probab. 16 (2011) 528–544.
  • [9] B. Ráth and A. Sapozhnikov. Connectivity properties of random interlacement and intersection of random walks. ALEA Lat. Am. J. Probab. Math. Stat. 9 (2012) 67–83.
  • [10] R. Schneider and W. Weil. Stochastic and Integral Geometry. Springer, Berlin, 2008.
  • [11] A.-S. Sznitman. Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 (3) (2010) 2039–2087.
  • [12] J. Tykesson and D. Windisch. Percolation in the vacant set of Poisson cylinders. Probab. Theory Related Fields 154 (2012) 165–191.