Percolation and contact processes with low-dimensional inhomogeneity
The inhomogeneous percolation model on \(\mathbb{Z}^d\), \(d\geq 2\), in consideration is defined as follows. Each bond in \(\{0\}^{d-s} \times \mathbb{Z}^s\) \((1\leq s\leq d-1)\) is open or closed with probability \(p_1\) or \(1-p_1\), respectively, and each remaining bond is open or closed with probability \(p_2\) or \(1-p_2\), respectively; all bonds of \(\mathbb{Z}^d\) behave independently. Let \(p_c (\mathbb{Z}^d)\) denote the critical value of \(p\) for the standard homogeneous (i.e. with \(p_1= p_2=p)\) percolation model on \(\mathbb{Z}^d\). The investigated problem is the appearance of percolation for the inhomogeneous model, with \(p_2= p_c (\mathbb{Z}^d)\) fixed, in dependence on \(p_1\). The relation of the studied problem to the study of the problem to show that there is no percolation at \(p=p_c (\mathbb{Z}^d)\), in the homogeneous problem, is explained. In fact, informations on the critical value \(p^s_c\) of \(p_1\) are deduced under assumptions stronger than that there is no percolation at \(p_c (\mathbb{Z}^d)\) for the homogeneous model. Related results for some inhomogeneous contact processes are presented. As a corollary, it is proved that (a) the percolation does not occur if \(s=1\), \(p_1\in[0,1)\), and \(d\) is sufficiently large; (b) there is a critical value \(p^s_c \in(p_c (\mathbb{Z}^d), p_c (\mathbb{Z}^s))\) for \(p_1\) if \(d\) is sufficiently large and \(2\leq s\leq d-3\). A related corollary for the contact processes is deduced under the assumption that an infrared bound holds for homogeneous contact processes if \(d\) is sufficiently large.
scientific article; zbMATH DE number 1130488