Annals of Probability
- Ann. Probab.
- Volume 34, Number 1 (2006), 181-218.
Random growth models with polygonal shapes
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.
Ann. Probab., Volume 34, Number 1 (2006), 181-218.
First available in Project Euclid: 17 February 2006
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 11N25: Distribution of integers with specified multiplicative constraints
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