Electronic Journal of Probability

Poisson cylinders in hyperbolic space

Erik Broman and Johan Tykesson

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We consider the Poisson cylinder model in $d$-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.


Article information

Electron. J. Probab., Volume 20 (2015), paper no. 41, 25 pp.

Accepted: 9 April 2015
First available in Project Euclid: 4 June 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Hyperbolic space continuum percolation

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Broman, Erik; Tykesson, Johan. Poisson cylinders in hyperbolic space. Electron. J. Probab. 20 (2015), paper no. 41, 25 pp. doi:10.1214/EJP.v20-3645. https://projecteuclid.org/euclid.ejp/1465067147

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  • Ahlfors, Lars V. Möbius transformations in several dimensions. Ordway Professorship Lectures in Mathematics. University of Minnesota, School of Mathematics, Minneapolis, Minn., 1981. ii+150 pp.
  • Beardon, Alan F. The geometry of discrete groups. Graduate Texts in Mathematics, 91. Springer-Verlag, New York, 1983. xii+337 pp. ISBN: 0-387-90788-2
  • Benjamini, Itai; Jonasson, Johan; Schramm, Oded; Tykesson, Johan. Visibility to infinity in the hyperbolic plane, despite obstacles. ALEA Lat. Am. J. Probab. Math. Stat. 6 (2009), 323–342.
  • Benjamini, Itai; Schramm, Oded. Percolation in the hyperbolic plane. J. Amer. Math. Soc. 14 (2001), no. 2, 487–507 (electronic).
  • Bollobás, Béla; Riordan, Oliver. The critical probability for random Voronoi percolation in the plane is 1/2. Probab. Theory Related Fields 136 (2006), no. 3, 417–468.
  • Broman, Erik; Tykesson, Johan. Connectedness of poisson cylinders in euclidean space, arXiv:1304.6357. To appear in Ann. Inst. H. Poincaré Probab. Statist.
  • Cannon, James W.; Floyd, William J.; Kenyon, Richard; Parry, Walter R. Hyperbolic geometry. Flavors of geometry, 59–115, Math. Sci. Res. Inst. Publ., 31, Cambridge Univ. Press, Cambridge, 1997.
  • Grimmett, Geoffrey. Percolation. Second edition. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 321. Springer-Verlag, Berlin, 1999. xiv+444 pp. ISBN: 3-540-64902-6
  • Häggström, Olle; Jonasson, Johan. Uniqueness and non-uniqueness in percolation theory. Probab. Surv. 3 (2006), 289–344.
  • Li, S. Concise formulas for the area and volume of a hyperspherical cap. Asian J. Math. Stat. 4 (2011), no. 1, 66–70.
  • Meester, Ronald; Roy, Rahul. Continuum percolation. Cambridge Tracts in Mathematics, 119. Cambridge University Press, Cambridge, 1996. x+238 pp. ISBN: 0-521-47504-X
  • Ratcliffe, John G. Foundations of hyperbolic manifolds. Second edition. Graduate Texts in Mathematics, 149. Springer, New York, 2006. xii+779 pp. ISBN: 978-0387-33197-3; 0-387-33197-2
  • Santaló, Luis A. Integral geometry and geometric probability. Second edition. With a foreword by Mark Kac. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2004. xx+404 pp. ISBN: 0-521-52344-3
  • Schneider, Rolf; Weil, Wolfgang. Stochastic and integral geometry. Probability and its Applications (New York). Springer-Verlag, Berlin, 2008. xii+693 pp. ISBN: 978-3-540-78858-4
  • Sznitman, Alain-Sol. Vacant set of random interlacements and percolation. Ann. of Math. (2) 171 (2010), no. 3, 2039–2087.
  • Teixeira, Augusto; Tykesson, Johan. Random interlacements and amenability. Ann. Appl. Probab. 23 (2013), no. 3, 923–956.
  • Tykesson, Johan; Windisch, David. Percolation in the vacant set of Poisson cylinders. Probab. Theory Related Fields 154 (2012), no. 1-2, 165–191.