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Asymptotic shape in a continuum growth model

Part of:Equilibrium statistical mechanicsSpecial 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: mia@math.su.se
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Abstract

A continuum growth model is introduced. The state at time t, S t , is a subset ofℝd and consists of a connected union of randomly sized Euclidean balls, which emerge from outbursts at their centre points. An outburst occurs somewhere in S t after an exponentially distributed time with expected value|S t |-1 and the location of the outburst is uniformly distributed over S t . The main result is that, if the distribution of the radii of the outburst balls has bounded support, then S t grows linearly and S t /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

Information

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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