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Stationary states of a directed planar growth process

Part of: Markov processes

Published online by Cambridge University Press:  01 July 2016

D. J. Gates
D. J. Gates*
Affiliation:
CSIRO
*
*Postal address: CSIRO Mathematical and Information Sciences, GPO Box 664, Canberra ACT 2601, Australia.
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Abstract

A new Markov process is introduced, describing growth or spread in two dimensions, via the aggregation of particles or the filling of cells. States of the process are configurations of part of the boundary of the growing aggregate, and transitions are captures or escapes of single particles. For suitably chosen transition rates, the process is dynamically reversible, leading to an explicit stationary distribution and a statistical description of the boundary. The growth rate is calculated and growth behaviour described. Different asymptotic relations between transition rates lead to different growth patterns or regimes. Besides the regimes familiar in polymer crystal growth, several new ones are described. The aggregate can have a porous structure resembling thin solid films deposited from vapour. Two measures of porosity, one for the boundary and one for the bulk, are calculated. The process is relevant to growing colonies of bacteria or the like, to the spread of epidemics and grass or forest fires, and to voter models.

Keywords

RICHARDSON MODELSPATIAL EPIDEMICCRYSTAL GROWTHVOTER MODELSDYNAMIC REVERSIBILITY

MSC classification

Primary: 60J20: Applications of Markov chains and discrete-time Markov processes on general state spaces (social mobility, learning theory, industrial processes, etc.)
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 4 , December 1997 , pp. 857 - 877
Copyright
Copyright © Probability Trust 1997 

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