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January 2006 Random growth models with polygonal shapes
Janko Gravner, David Griffeath
Ann. Probab. 34(1): 181-218 (January 2006). DOI: 10.1214/009117905000000512


We consider discrete-time random perturbations of monotone cellular automata (CA) in two dimensions. Under general conditions, we prove the existence of half-space velocities, and then establish the validity of the Wulff construction for asymptotic shapes arising from finite initial seeds. Such a shape converges to the polygonal invariant shape of the corresponding deterministic model as the perturbation decreases. In many cases, exact stability is observed. That is, for small perturbations, the shapes of the deterministic and random processes agree exactly. We give a complete characterization of such cases, and show that they are prevalent among threshold growth CA with box neighborhood. We also design a nontrivial family of CA in which the shape is exactly computable for all values of its probability parameter.


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Janko Gravner. David Griffeath. "Random growth models with polygonal shapes." Ann. Probab. 34 (1) 181 - 218, January 2006.


Published: January 2006
First available in Project Euclid: 17 February 2006

zbMATH: 1090.60077
MathSciNet: MR2206346
Digital Object Identifier: 10.1214/009117905000000512

Primary: 60K35
Secondary: 11N25

Keywords: asymptotic shape , cellular automaton , exact stability , Growth model

Rights: Copyright © 2006 Institute of Mathematical Statistics

Vol.34 • No. 1 • January 2006
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