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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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