dorsal/arxiv
View SchemaMetastability in the Greenberg-Hastings Model
| Authors | Robert Fisch, Janko Gravner, David Griffeath |
|---|---|
| Categories | |
| ArXiv ID | patt-sol/9303005 |
| URL | https://arxiv.org/abs/patt-sol/9303005 |
Abstract
The Greenberg-Hastings Model (GHM) is a family of multitype cellular automata that emulate excitable media, exhibiting the nucleation and spiral formation characteristic of such complex systems. In this paper we study the asymptotic frequency of nucleation in 2-d GHM dynamics as the number k of types, or colors, becomes large. Starting from uniform product measure over colors, and assuming that the excitation threshold is not too large, the box size needed for formation of a spiral core is shown to grow exponentially in k. By exploiting connections with percolation theory we find that a box of size exp{(.23+-.06)k} is required in the nearest neighbor, threshold 1 case. In contrast, GHM rules obey power law nucleation scaling when started from a suitable non-uniform product measure over the k colors; this effect is driven by critical percolation. Finally, we present some analogous results for a Random GHM, an interacting Markovian system closely related to the epidemic with regrowth of Durrett and Neuhauser.
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"abstract": "The Greenberg-Hastings Model (GHM) is a family of multitype cellular automata\nthat emulate excitable media, exhibiting the nucleation and spiral formation\ncharacteristic of such complex systems. In this paper we study the asymptotic\nfrequency of nucleation in 2-d GHM dynamics as the number k of types, or\ncolors, becomes large. Starting from uniform product measure over colors, and\nassuming that the excitation threshold is not too large, the box size needed\nfor formation of a spiral core is shown to grow exponentially in k. By\nexploiting connections with percolation theory we find that a box of size\nexp{(.23+-.06)k} is required in the nearest neighbor, threshold 1 case. In\ncontrast, GHM rules obey power law nucleation scaling when started from a\nsuitable non-uniform product measure over the k colors; this effect is driven\nby critical percolation. Finally, we present some analogous results for a\nRandom GHM, an interacting Markovian system closely related to the epidemic\nwith regrowth of Durrett and Neuhauser.",
"arxiv_id": "patt-sol/9303005",
"authors": [
"Robert Fisch",
"Janko Gravner",
"David Griffeath"
],
"categories": [
"patt-sol",
"adap-org",
"nlin.AO",
"nlin.PS"
],
"title": "Metastability in the Greenberg-Hastings Model",
"url": "https://arxiv.org/abs/patt-sol/9303005"
},
"schema_id": "dorsal/arxiv",
"source": {
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"variant": "snapshot-2026-03-01",
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