Threshold growth dynamics

Authors:
Janko Gravner and David Griffeath

Journal:
Trans. Amer. Math. Soc. **340** (1993), 837-870

MSC:
Primary 52A37; Secondary 60K35

MathSciNet review:
1147400

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Abstract: We study the asymptotic shape of the occupied region for monotone deterministic dynamics in -dimensional Euclidean space parametrized by a *threshold* , and a Borel set with positive and finite Lebesgue measure. If denotes the oocupied set of the dynamics at integer time , then is obtained by adjoining any point for which the volume of overlap between and exceeds . Except in some degenerate cases, we prove that converges to a unique limiting "shape" starting from any bounded initial region that is suitably large. Moreover, is computed as the polar transform for , where is an explicit *width* function that depends on and .

It is further shown that describes the limiting shape of wave fronts for certain cellular automaton growth rules related to lattice models of excitable media, as the threshold and range of interaction increase suitably. In the case of *box* neighborhoods on , these limiting shapes are calculated and the dependence of their anisotropy on is examined. Other specific two- and three-dimensional examples are also discussed in some detail.

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1993-1147400-3

Keywords:
Growth model,
threshold dynamics,
excitable medium,
polar transform

Article copyright:
© Copyright 1993
American Mathematical Society