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March 2013 Full- and half-Gilbert tessellations with rectangular cells
James Burridge, Richard Cowan, Isaac Ma
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Adv. in Appl. Probab. 45(1): 1-19 (March 2013). DOI: 10.1239/aap/1363354100


We investigate the ray-length distributions for two different rectangular versions of Gilbert's tessellation (see Gilbert (1967)). In the full rectangular version, lines extend either horizontally (east- and west-growing rays) or vertically (north- and south-growing rays) from seed points which form a Poisson point process, each ray stopping when another ray is met. In the half rectangular version, east- and south-growing rays do not interact with west and north rays. For the half rectangular tessellation, we compute analytically, via recursion, a series expansion for the ray-length distribution, whilst, for the full rectangular version, we develop an accurate simulation technique, based in part on the stopping-set theory for Poisson processes (see Zuyev (1999)), to accomplish the same. We demonstrate the remarkable fact that plots of the two distributions appear to be identical when the intensity of seeds in the half model is twice that in the full model. In this paper we explore this coincidence, mindful of the fact that, for one model, our results are from a simulation (with inherent sampling error). We go on to develop further analytic theory for the half-Gilbert model using stopping-set ideas once again, with some novel features. Using our theory, we obtain exact expressions for the first and second moments of the ray length in the half-Gilbert model. For all practical purposes, these results can be applied to the full-Gilbert model—as much better approximations than those provided by Mackisack and Miles (1996).


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James Burridge. Richard Cowan. Isaac Ma. "Full- and half-Gilbert tessellations with rectangular cells." Adv. in Appl. Probab. 45 (1) 1 - 19, March 2013.


Published: March 2013
First available in Project Euclid: 15 March 2013

zbMATH: 1281.60011
MathSciNet: MR3077538
Digital Object Identifier: 10.1239/aap/1363354100

Primary: 05B45 , 60D05
Secondary: 51M20 , 52C17 , 60G55

Keywords: crack formation , division of space , fragmentation , point process , random tessellation

Rights: Copyright © 2013 Applied Probability Trust


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Vol.45 • No. 1 • March 2013
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