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A note on the lilypond model

Part of: Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Codina Cotar  and
Stanislav Volkov
Codina Cotar*
Affiliation:
University of Bristol
Stanislav Volkov*
Affiliation:
University of Bristol
*
∗ Postal address: Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK
∗ Postal address: Department of Mathematics, University of Bristol, Bristol BS8 1TW, UK
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Abstract

We consider some generalizations of the germ-grain growing model studied by Daley, Mallows and Shepp (2000). In this model, a realization of a Poisson process on a line with points Xi is fixed. At time zero, simultaneously at each Xi, a circle (grain) starts growing at the same speed. It grows until it touches another grain, and then it stops. The question is whether the point zero is eventually covered by some circle. In our note we expand this model in the following three directions. We study: a one-sided growth model with a fixed number of circles; a grain-growth model on a regular tree; and a grain-growth model on a line with non-Poisson distributed centres of the circles.

Keywords

Lilypond modelPoisson point processnon-overlapping spheresgerm-growth model

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 36 , Issue 2 , June 2004 , pp. 325 - 339
Copyright
Copyright © Applied Probability Trust 2004 

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