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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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