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Random growth in a tessellation

Published online by Cambridge University Press:  24 October 2008

Daniel Richardson
Affiliation:
Department of Mathematics, Polytechnic of the South Bank, Borough Road, London S.E.1

Extract

Let S be n dimensional Euclidean space and let T be a division of S into cells. Assume that each cell must be either white or black at any time t. At time 0 the cell at the origin, α0, is black and all other cells are white. Let G be some stochastic growth process which tends to change white cells with black neighbours into black cells. Let C(t) be the black shape at time t. For a family, F, of such growth processes we prove the following theorem.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 1973

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References

REFERENCES

(1)Coxeter, H. S. M.Regular polytopes (London, Methuen, 1948).Google Scholar
(2)Eden, M. A two-dimensional growth process. Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability. Vol. IV. Biology and problems of health, ed. Neyman, J. (1961).Google Scholar
(3)Hammersley, J. M.First passage percolation, J. Roy. Statist. Soc. 28, 1966.Google Scholar
(4)Hammersley, J. M. and Welsh, D. J. First passage percolation, subadditive processes, stochastic networks, and generalized renewal theory, in Bernoulli (1723), Bayes (1763), Laplace (1813) (Springer-Verlag, 1965).Google Scholar
(5)Morogan, R. W. and Welsh, D. J.A two-dimensional Poisson growth process. J. Roy. Statist. Soc. 27, 1965.Google Scholar
(6)Williams, T. and Bjerknes, R.A stochastic model for abnormal clone spread through epithelial basal layer. Nature 236, 3 03 1972.CrossRefGoogle ScholarPubMed