Aggregate and fractal tessellations

Konstantin Tchoumatchenko, Sergei Zuev

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    Abstract

    Consider a sequence of stationary tessellations {‹n}, n=0,1,..., of  d consisting of cells {Cn(xin)}with the nuclei {xin}. An aggregate cell of level one, C01(xi0), is the result of merging the cells of ‹1 whose nuclei lie in C0(xi0). An aggregate tessellation ‹0n consists of the aggregate cells of level n, C0n(xi0), defined recursively by merging those cells of ‹n whose nuclei lie in Cnm1(xi0). We find an expression for the probability for a point to belong to atypical aggregate cell, and obtain bounds for the rate of itsexpansion. We give necessary conditions for the limittessellation to exist as nMX and provide upperbounds for the Hausdorff dimension of its fractal boundary and forthe spherical contact distribution function in the case ofPoisson-Voronoi tessellations {‹n}.
    Original languageEnglish
    Pages (from-to)198-218
    Number of pages20
    JournalProbability Theory and Related Fields
    Volume121
    Issue number2
    DOIs
    Publication statusPublished - 31 Oct 2001

    Keywords

    • probability
    • voronoï tessellation
    • pProbability distribution
    • telecommunications
    • poisson process
    • fractal

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