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Asymptotic shape in a continuum growth model

Part of: Equilibrium statistical mechanics Special processes

Published online by Cambridge University Press:  22 February 2016

Maria Deijfen
Maria Deijfen*
Affiliation:
Stockholm University
*
∗ Postal address: Department of Mathematics, Stockholm University, SE-106 91 Stockholm, Sweden. Email address: [email protected]
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Abstract

A continuum growth model is introduced. The state at time t, St, is a subset of ℝd and consists of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their centre points. An outburst occurs somewhere in St after an exponentially distributed time with expected value |St|-1 and the location of the outburst is uniformly distributed over St. The main result is that, if the distribution of the radii of the outburst balls has bounded support, then St grows linearly and St/t has a nonrandom shape as t → ∞. Due to rotational invariance the asymptotic shape must be a Euclidean ball.

Keywords

First passage percolationRichardson's modelsubadditivityshape theorem

MSC classification

Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory
Secondary: 82B43: Percolation
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 35 , Issue 2 , June 2003 , pp. 303 - 318
Copyright
Copyright © Applied Probability Trust 2003 

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References

Boivin, D. (1990). First passage percolation: the stationary case. Prob. Theory Relat. Fields 86, 491–499.CrossRefGoogle Scholar
Bramson, M. and Griffeath, D. (1981). On the Williams–Bjerknes tumour growth model I. Ann. Prob. 9, 173–185.Google Scholar
Cox, J. T. and Durrett, R. (1981). Some limit theorems for percolation processes with necessary and sufficient conditions. Ann. Prob. 9, 583–603.Google Scholar
Eden, M. (1961). A two-dimensional growth process. In Proc. 4th Berkeley Symp. Math. Statist. Prob., Vol. IV, University of California Press, Berkeley, pp. 223–239.Google Scholar
Howard, C. D. and Newman, C. M. (1997). Euclidean models of first-passage percolation. Prob. Theory Relat. Fields 108, 153–170.CrossRefGoogle Scholar
Kesten, H. (1986). Aspects of first-passage percolation. In École d'été de probabilités de Saint-Flour, 1984 (Lecture Notes Math. 1180), Springer, Berlin, pp. 125–264.Google Scholar
Lee, T. and Cowan, R. (1994). A stochastic tessellation of digital space. In Mathematical Morphology and Its Applications to Image Processing, ed. Serra, J., Kluwer, Dordrecht, pp. 217–224.Google Scholar
Liggett, T. M. (1985). An improved subadditive ergodic theorem. Ann. Prob. 13, 1279–1285.CrossRefGoogle Scholar
Penrose, M. (2001). Random parking, sequential adsorption and the jamming limit. Commun. Math. Phys. 218, 153–176.Google Scholar
Richardson, D. (1973). Random growth in a tessellation. Proc. Camb. Phil. Soc. 74, 515–528.Google Scholar
Williams, D. (1991). Probability with Martingales. Cambridge University Press.Google Scholar
Williams, T. and Bjerknes, R. (1972). Stochastic model for abnormal clone spread through epithelial basal layer. Nature 236, 19–21.CrossRefGoogle ScholarPubMed