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Interaction Processes for Unions of Facets, the Asymptotic Behaviour with Increasing Intensity

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  • Jakub Večeřa

    (Charles University in Prague, Faculty of Mathematics and Physics)

  • Viktor Beneš

    (Charles University in Prague, Faculty of Mathematics and Physics)

Abstract

In the series of models with interacting particles in stochastic geometry, a further contribution presents the facet process which is defined in arbitrary Euclidean dimension. In 2D, 3D specially it is a process of interacting segments, flat surfaces, respectively. Its investigation is based on the theory of functionals of finite spatial point processes given by a density with respect to a Poisson process. The methodology based on L 2 expansion of the covariance of functionals of Poisson process is developed for U-statistics of facet intersections which are building blocks of the model. The importance of the concept of correlation functions of arbitrary order is emphasized. Some basic properties of facet processes, such as local stability and repulsivness are shown and a standard simulation algorithm mentioned. Further the situation when the intensity of the process tends to infinity is studied. In the case of Poisson processes a central limit theorem follows from recent results of Wiener-Ito theory. In the case of non-Poisson processes we restrict to models with finitely many orientations. Detailed analysis of correlation functions exhibits various asymptotics for different combination of U-statistics and submodels of the facet process.

Suggested Citation

  • Jakub Večeřa & Viktor Beneš, 2016. "Interaction Processes for Unions of Facets, the Asymptotic Behaviour with Increasing Intensity," Methodology and Computing in Applied Probability, Springer, vol. 18(4), pages 1217-1239, December.
    • Handle: RePEc:spr:metcap:v:18:y:2016:i:4:d:10.1007_s11009-016-9485-8
    DOI: 10.1007/s11009-016-9485-8
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    References listed on IDEAS

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    1. Zbyněk Pawlas, 2014. "Self-crossing Points of a Line Segment Process," Methodology and Computing in Applied Probability, Springer, vol. 16(2), pages 295-309, June.
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    Cited by:

    1. Viktor Beneš & Jakub Večeřa & Milan Pultar, 2019. "Planar Segment Processes with Reference Mark Distributions, Modeling and Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 683-698, September.

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    1. Viktor Beneš & Jakub Večeřa & Milan Pultar, 2019. "Planar Segment Processes with Reference Mark Distributions, Modeling and Estimation," Methodology and Computing in Applied Probability, Springer, vol. 21(3), pages 683-698, September.

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