Planar tessellations that have the half-Gilbert structure

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Abstract

In the full rectangular version of Gilbert's planar tessellation (see Gilbert (1967), Mackisack and Miles (1996), and Burridge et al. (2013)), lines extend either horizontally (with east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a stationary Poisson point process, each ray stopping when it meets another ray that has blocked its path. In the half-Gilbert rectangular version, east- and south-growing rays, whilst having the potential to block each other, do not interact with west and north rays, and vice versa. East- and south-growing rays have a reciprocality of blocking, as do west and north. In this paper we significantly expand and simplify the half-Gilbert analytic results that we gave in Burridge et al. (2013). We also show how the idea of reciprocality of blocking between rays can be used in a much wider context, with rays not necessarily orthogonal and with seeds that produce more than two rays.
Original languageEnglish
Pages (from-to)574-584
Number of pages11
JournalAdvances in Applied Probability
Volume48
Issue number2
DOIs
Publication statusPublished - 10 Jun 2016

Keywords

  • random tessellation
  • point process
  • crack formation
  • division of space

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