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Large-range constant threshold growth model in one dimension. (English) Zbl 1189.60185
Summary: We study a one dimensional constant threshold model in continuous time. Its dynamics have two parameters, the range $$n$$ and the threshold $$v$$. An unoccupied site $$x$$ becomes occupied at rate $$1$$ as soon as there are at least $$v$$ occupied sites in $$[x-n, x+n]$$. As $$n$$ goes to infinity and $$v$$ is kept fixed, the dynamics can be approximated by a continuous space version, which has an explicit invariant measure at the front. This allows us to prove that the speed of propagation is asymptotically $$n^{2}/2v$$.
##### MSC:
 60K35 Interacting random processes; statistical mechanics type models; percolation theory 82C22 Interacting particle systems in time-dependent statistical mechanics
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