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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.)
##### Keywords:
branching particle processes; contact process
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