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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Discrete threshold growth dynamics are omnivorous for box neighborhoods
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by Tom Bohman PDF
Trans. Amer. Math. Soc. 351 (1999), 947-983 Request permission

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
  • 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