Critical intensities of Boolean models with different underlying convex shapes

Rahul Roy, Hideki Tanemura

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

We consider the Poisson Boolean model of percolation where the percolating shapes are convex regions. By an enhancement argument we strengthen a result of Jonasson (2001) to show that the critical intensity of percolation in two dimensions is minimized among the class of convex shapes of unit area when the percolating shapes are triangles, and, for any other shape, the critical intensity is strictly larger than this minimum value. We also obtain a partial generalization to higher dimensions. In particular, for three dimensions, the critical intensity of percolation is minimized among the class of regular polytopes of unit volume when the percolating shapes are tetrahedrons. Moreover, for any other regular polytope, the critical intensity is strictly larger than this minimum value.

Original languageEnglish
Pages (from-to)48-57
Number of pages10
JournalAdvances in Applied Probability
Volume34
Issue number1
DOIs
Publication statusPublished - 2002 Mar 1
Externally publishedYes

Keywords

  • Boolean model
  • Critical intensity
  • Percolation
  • Poisson process

ASJC Scopus subject areas

  • Statistics and Probability
  • Applied Mathematics

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