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Discrete threshold growth dynamics are omnivorous for box neighborhoods

Authors:	Tom Bohman

Institution:	Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903

Abstract:	In the discrete threshold model for crystal growth in the plane we begin with some set $A_{0} \subset {\mathbf Z}^{2}$ of seed crystals and observe crystal growth over time by generating a sequence of subsets $A_{0} \subset A_{1} \subset A_{2} \subset \dotsb$ of ${\mathbf Z}^{2}$ by a deterministic rule. This rule is as follows: a site crystallizes when a threshold number of crystallized points appear in the site's prescribed neighborhood. The growth dynamics generated by this model are said to be omnivorous if $A_{0}$ finite and $A_{i+1} \neq A_{i} \forall i$ imply $\bigcup _{i=0}^{\infty} A_{i} = {\mathbf Z}^{2}$ . In this paper we prove that the dynamics are omnivorous when the neighborhood is a box (i.e. when, for some fixed $\rho$ , the neighborhood of is $\{ x \in {\mathbf Z}^{2} : \\|x-z\\|_{\infty} \le \rho\})$ . This result has important implications in the study of the first passage time when $A_{0}$ is chosen randomly with a sparse Bernoulli density and in the study of the limiting shape to which $n^{-1}A_{n}$ converges.

Keywords:	Threshold growth cellular automata

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