Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-26T05:55:41.234Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Çinlar, E., (1975). Introduction to Stochastic Processes. Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Daley, D. J., Mallows, C. L. and Shepp, L. A. (2000). A one-dimensional Poisson growth model with non-overlapping intervals. Stoch. Process. Appl. 90, 223–241.Google Scholar
Daley, D. J., Stoyan, H. and Stoyan, D. (1999). The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains. Adv. Appl. Prob. 31, 610–624.Google Scholar
Feller, W. (1971). An Introduction to Probability Theory and Its Applications, Vol. 2, 2nd edn. John Wiley, New York.Google Scholar
Häggeström, O. and Meester, R. (1996). Nearest neighbor and hard sphere models in continuum percolation. Random Structures Algorithms 9, 295–315.Google Scholar
Huffer, F. W. (2002). One-dimensional Poisson growth models with random and asymmetric growth. Methodology Comput. Appl. Prob. 4, 257–278.Google Scholar
Istratescu, V. I. (1981). Fixed Point Theory. Reidel, Dordrecht.Google Scholar
Kingman, J. F. C. (1972). Regenerative Phenomena. John Wiley, London.Google Scholar
Kolmogorov, A. N. and Fomin, S.V. (1957). Elements of the Theory of Functions and Functional Analysis. Graylock, Rochester, NY.Google Scholar
O'Hely, M. and Sudbury, A. W. (2001). The annihilating process. J. Appl. Prob. 38, 223–231.Google Scholar
Sudbury, A. (2002). Inclusion–exclusion methods for treating annihilating and deposition processes. J. Appl. Prob. 39, 466–478.Google Scholar