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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Discrete threshold growth dynamics are omnivorous for box neighborhoods

Author: Tom Bohman
Journal: Trans. Amer. Math. Soc. 351 (1999), 947-983
MSC (1991): Primary 60K35; Secondary 05D99
MathSciNet review: 1443863
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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 $z$ 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.

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Additional Information

Tom Bohman
Affiliation: Department of Mathematics, Rutgers University, New Brunswick, New Jersey 08903
Address at time of publication: Department of Mathematics, 2-339, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Keywords: Threshold growth, cellular automata
Received by editor(s): August 19, 1996
Received by editor(s) in revised form: February 7, 1997
Article copyright: © Copyright 1999 American Mathematical Society