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Statistical mechanics of crabgrass. (English) Zbl 0682.60090
The authors investigate the time evolution of the contact branching particle processes on \({\mathbb{Z}}^ d/M=\{(x_ 1/M,...,x_ d/M)\}\in {\mathbb{R}}^ d,\) \(x\in {\mathbb{Z}}^ d\). The critical rate of births \(\lambda_ c(M)>0\) is introduced, \(\lambda_ c(M)=\inf \{\lambda:\) the contact process survives with positive probability\(\}\). It is shown that \(\lambda_ c(M)\to 1\) as \(M\to \infty\), and the rate of convergence depends upon the dimension: \(\lambda_ c(M)-1\approx M^{-2/3}\) for \(d=1\); \(\approx (\ln M)/M^ 2\) for \(d=2\); and \(M^{-d}\) for \(d\geq 3\).
Reviewer: V.Chulaevski

MSC:
60K35 Interacting random processes; statistical mechanics type models; percolation theory
60J80 Branching processes (Galton-Watson, birth-and-death, etc.)
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