Annals of Probability

Random growth models with polygonal shapes

Janko Gravner and David Griffeath

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Abstract

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.

Article information

Source
Ann. Probab., Volume 34, Number 1 (2006), 181-218.

Dates
First available in Project Euclid: 17 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.aop/1140191536

Digital Object Identifier
doi:10.1214/009117905000000512

Mathematical Reviews number (MathSciNet)
MR2206346

Zentralblatt MATH identifier
1090.60077

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 11N25: Distribution of integers with specified multiplicative constraints

Keywords
Cellular automaton growth model asymptotic shape exact stability

Citation

Gravner, Janko; Griffeath, David. Random growth models with polygonal shapes. Ann. Probab. 34 (2006), no. 1, 181--218. doi:10.1214/009117905000000512. https://projecteuclid.org/euclid.aop/1140191536


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