文章基本信息

标题：Covering the whole space with Poisson random balls
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作者：Hermine Biermé ; Anne Estrade
期刊名称：Latin American Journal of Probability and Mathematical Statistics
电子版ISSN：1980-0436
出版年度：2012
卷号：IX
页码：213-229
出版社：Instituto Nacional De Matemática Pura E Aplicada
摘要：We consider Poisson random balls in Rd, with the pair (center, radius)given by a Poisson point process in Rd×(0,+1). According to the intensity measureof the Poisson process, we investigate the eventuality of covering the whole spacewith the union of the balls. We exhibit a disjunction phenomenon between thecoverage with large balls (low frequency) and the coverage with small balls (highfrequency). Concerning the second type of coverage, we prove the existence of acritical regime which separates the case where coverage occurs almost surely andthe case where coverage does not occur almost surely. We give an explicit value ofthe critical intensity and we prove that the Hausdorff measure of the set of pointswhich are not covered by the union of balls is linked with this value. We alsocompare with other critical regimes appearing in continuum percolation.
关键词：Poisson point process; coverage; random set; Boolean model; perco-;lation; Hausdorff dimension.