dorsal/arxiv
View SchemaThreshold Growth Dynamics
| Authors | Janko Gravner, David Griffeath |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9303004 |
| URL | https://arxiv.org/abs/patt-sol/9303004 |
Abstract
We study the asymptotic shape of the occupied region for monotone deterministic dynamics in d-dimensional Euclidean space parametrized by a threshold theta, and a Borel set N with positive and finite Lebesgue measure. If A_n denotes the occupied set of the dynamics at integer time n, then A_n+1 is obtained by adjoining any point x for which the volume of overlap between x+N and A_n exceeds theta. Except in some degenerate cases, we prove that A_n converges to a unique limiting "shape" L starting from any bounded initial region that is suitably large. Moreover, L is computed as the polar transform for 1/w, where w is an explicit width function that depends on N and theta. It is further shown that L 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 2-d box neighborhoods, these limiting shapes are calculated and the dependence of their anisotropy on theta is examined. Other specific two- and three- dimensional examples are also discussed in some detail.
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"abstract": "We study the asymptotic shape of the occupied region for monotone\ndeterministic dynamics in d-dimensional Euclidean space parametrized by a\nthreshold theta, and a Borel set N with positive and finite Lebesgue measure.\nIf A_n denotes the occupied set of the dynamics at integer time n, then A_n+1\nis obtained by adjoining any point x for which the volume of overlap between\nx+N and A_n exceeds theta. Except in some degenerate cases, we prove that A_n\nconverges to a unique limiting \"shape\" L starting from any bounded initial\nregion that is suitably large. Moreover, L is computed as the polar transform\nfor 1/w, where w is an explicit width function that depends on N and theta. It\nis further shown that L describes the limiting shape of wave fronts for certain\ncellular automaton growth rules related to lattice models of excitable media,\nas the threshold and range of interaction increase suitably. In the case of 2-d\nbox neighborhoods, these limiting shapes are calculated and the dependence of\ntheir anisotropy on theta is examined. Other specific two- and three-\ndimensional examples are also discussed in some detail.",
"arxiv_id": "patt-sol/9303004",
"authors": [
"Janko Gravner",
"David Griffeath"
],
"categories": [
"patt-sol",
"adap-org",
"nlin.AO",
"nlin.PS"
],
"title": "Threshold Growth Dynamics",
"url": "https://arxiv.org/abs/patt-sol/9303004"
},
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