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An explicit expression for the distribution of the number of sides of the typical Poisson-Voronoi cell

Part of: Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Pierre Calka
Pierre Calka*
Affiliation:
Université Claude Bernard Lyon 1
*
∗ Postal address: Université Claude Bernard Lyon 1, LaPCS, Bât. B, Domaine de Gerland, 50, avenue Tony-Garnier, F-69366 Lyon Cedex 07, France. Email address: [email protected]
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Abstract

In this paper, we give an explicit expression for the distribution of the number of sides (or equivalently vertices) of the typical cell of a two-dimensional Poisson-Voronoi tessellation. We use this formula to give a table of numerical values of the distribution function.

Keywords

Palm distributionPoisson-Voronoi tessellationstochastic geometrytypical cell

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 4 , December 2003 , pp. 863 - 870
Copyright
Copyright © Applied Probability Trust 2003 

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References

[1] Baccelli, F. and Błaszczyszyn, B. (2001). On a coverage process ranging from the Boolean model to the Poisson–Voronoi tessellation with applications to wireless communications. Adv. Appl. Prob. 33, 293–323.Google Scholar
[2] Calka, P. (2002). The distributions of the smallest disks containing the Poisson–Voronoi typical cell and the Crofton cell in the plane. Adv. Appl. Prob. 34, 702–717.Google Scholar
[3] Calka, P. (2003). Precise formulae for the distributions of the principal geometric characteristics of the typical cells of a two-dimensional Poisson–Voronoi tessellation and a Poisson line process. Adv. Appl. Prob. 35, 551–562.Google Scholar
[4] Gilbert, E. N. (1962). Random subdivisions of space into crystals. Ann. Math. Statist. 33, 958–972.CrossRefGoogle Scholar
[5] Goldman, A. and Calka, P. (2003). On the spectral function of the Poisson–Voronoi cells. To appear in Ann. Inst. H. Poincaré Prob. Statist.Google Scholar
[6] Hayen, A. and Quine, M. (2000). Calculating the proportion of triangles in a Poisson–Voronoi tessellation of the plane. J. Statist. Comput. Simul. 67, 351–358.CrossRefGoogle Scholar
[7] Hayen, A. and Quine, M. (2000). The proportion of triangles in a Poisson–Voronoi tessellation of the plane. Adv. Appl. Prob. 32, 67–74.Google Scholar
[8] Hinde, A. L. and Miles, R. E. (1980). Monte-Carlo estimates of the distributions of the random polygons of the Voronoi tessellation with respect to a Poisson process. J. Statist. Comput. Simul. 10, 205–223.Google Scholar
[9] Kumar, S. and Kurtz, S. K. (1993). Properties of a two-dimensional Poisson–Voronoi tessellation: a Monte-Carlo study. Materials Characterization 31, 55–68.Google Scholar
[10] Kumar, S. and Singh, R. N. (1995). Thermal conductivity of polycristalline materials. J. Amer. Cer. Soc. 78, 728–736.CrossRefGoogle Scholar
[11] Le Caër, G. and Ho, J. S. (1990). The Voronoi tessellation generated from eigenvalues of complex random matrices. J. Phys. A 23, 3279–3295.CrossRefGoogle Scholar
[12] Meijering, J. L. (1953). Interface area, edge length, and number of vertices in crystal aggregates with random nucleation. Philips Res. Rep. 8, 270–290.Google Scholar
[13] Miles, R. E. and Maillardet, R. J. (1982). The basic structures of Voronoi and generalized Voronoi polygons. In Essays In Statistical Science (J. Appl. Prob. Spec. Vol. 19A), eds Gani, J. and Hannan, E. J., Applied Probability Trust, Sheffield, pp. 97–111.Google Scholar
[14] Möller, J., (1994). Lectures on Random Voronoi Tessellations. Springer, New York.Google Scholar
[15] Okabe, A., Boots, B., Sugihara, K. and Chiu, S. N. (2000). Spatial Tessellations: Concepts and Applications of Voronoi Diagrams, 2nd edn. John Wiley, Chichester.CrossRefGoogle Scholar
[16] Pielou, E. (1977). Mathematical Ecology. John Wiley, New York.Google Scholar
[17] Stoyan, D., Kendall, W. S. and Mecke, J. (1987). Stochastic Geometry and Its Applications. John Wiley, Chichester.Google Scholar
[18] Van de Weygaert, R. (1994). Fragmenting the Universe III. The construction and statistics of 3-D Voronoi tessellations. Astron. Astrophys. 283, 361–406.Google Scholar
[19] Zuyev, S. A. (1992). Estimates for distributions of the Voronoi polygon's geometric characteristics. Random Structures Algorithms 3, 149–162.Google Scholar