Electronic Journal of Probability

Shape theorem and surface fluctuation for Poisson cylinders

Marcelo Hilario, Xinyi Li, and Petr Panov

Full-text: Open access

Abstract

We prove a shape theorem for Poisson cylinders, and give a power-law bound on surface fluctuations. In particular, we show that for any $a \in (1/2, 1)$, conditioned on the origin being in the set of cylinders, if a point belongs to this set and has Euclidean norm below $R$, then this point lies at internal distance less than $R + O(R^{a})$ from the origin.

Article information

Source
Electron. J. Probab., Volume 24 (2019), paper no. 68, 16 pp.

Dates
Received: 27 July 2018
Accepted: 28 May 2019
First available in Project Euclid: 28 June 2019

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1561687601

Digital Object Identifier
doi:10.1214/19-EJP329

Mathematical Reviews number (MathSciNet)
MR3978218

Zentralblatt MATH identifier
07089006

Subjects
Primary: 60F10: Large deviations 82B43: Percolation [See also 60K35] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 51F99: None of the above, but in this section

Keywords
Poisson cylinder model internal distance shape theorem

Rights
Creative Commons Attribution 4.0 International License.

Citation

Hilario, Marcelo; Li, Xinyi; Panov, Petr. Shape theorem and surface fluctuation for Poisson cylinders. Electron. J. Probab. 24 (2019), paper no. 68, 16 pp. doi:10.1214/19-EJP329. https://projecteuclid.org/euclid.ejp/1561687601


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References

  • [1] D. J. Aldous and W. S. Kendall. Short-length routes in low-cost networks via Poisson line patterns. Adv. Appl. Probab.. 40: 1-21, 2008.
  • [2] A. Auffinger, M. Damron and J. Hanson. 50 years of first passage percolation. AMS University Lecture Series, 68, 2017.
  • [3] E. I. Broman and J. Tykesson. Connectedness of Poisson cylinders in Euclidean space. Ann. Inst. H. Poincaré Probab. Statist., 52(1): 102-126, 2016.
  • [4] E. I. Broman and J. Tykesson. Poisson cylinders in hyperbolic space. Electron. J. Probab., 20(41): 1-25, 2015.
  • [5] J. Černý and S. Popov. On the internal distance in the interlacement set. Electron. J. Probab., 17(29): 1-25, 2012.
  • [6] A. Drewitz, B. Ráth and A. Sapozhnikov. On chemical distances and shape theorems in percolation models with long-range correlations. J. Math. Phys., 55(8): 083307, 2014.
  • [7] M. R. Hilario, V. Sidoravicius and A. Teixeira. Cylinders’ percolation in three dimensions. Probab. Theory Relat. Fields, 163(3-4): 613-642, 2015.
  • [8] W. S. Kendall. Geodesics and flows in a Poissonian city. Ann. Appl. Probab., 21(3): 801-842, 2011.
  • [9] H. Kesten. On the speed of Convergence in First-Passage Percolation. Ann. Appl. Probab., 3(2): 296-338, 1993.
  • [10] X. Li. Percolative properties of Brownian interlacements and its vacant set. To appear in J. Theor. Probab., also available at arXiv:1610.08204.
  • [11] R. E. Miles. Poisson flats in Euclidean spaces. Part I: A finite number of random uniform flats. Adv. Appl. Probab., 1(2): 211-237, 1969.
  • [12] B. Ráth and A. Sapozhnikov. On the transience of random interlacements. Electron. Commun. Probab., 16(35): 379-391, 2011.
  • [13] B. Ráth and A. Sapozhnikov. Connectivity properties of random interlacement and intersection of random walks. ALEA, Lat. Am. Probab. Math. Stat., 9: 67-83, 2012.
  • [14] M. Spiess. Characteristics of Poisson cylinder processes and their estimation. Dissertation. Open Access Repositorium der Universität Ulm, OPARU-2538, 2012.
  • [15] J. Tykesson and D. Windisch. Percolation in the vacant set of Poisson cylinders. Probab. Theory Relat. Fields, 154(1-2): 165-191, 2012.