## Discrete threshold growth dynamics are omnivorous for box neighborhoods

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- by Tom Bohman
- Trans. Amer. Math. Soc.
**351**(1999), 947-983 - DOI: https://doi.org/10.1090/S0002-9947-99-02018-8
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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.## References

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## Bibliographic 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
- Email: bohman@math.mit.edu
- Received by editor(s): August 19, 1996
- Received by editor(s) in revised form: February 7, 1997
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 947-983 - MSC (1991): Primary 60K35; Secondary 05D99
- DOI: https://doi.org/10.1090/S0002-9947-99-02018-8
- MathSciNet review: 1443863