Statistical mechanics of crabgrass (Q1824292)

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 \(d\geq 3\).