Percolation and contact processes with low-dimensional inhomogeneity

Abstract

We consider inhomogeneous nearest neighbor Bernoulli bond percolation on $mathbb{Z}^d$ where the bonds in a fixed $s$-dimensional hyperplane $1\leq s\leq d-1)$ have density $p_1$ and all other bonds have fixed density, $p_c(\mathbb{Z}^d)$, the homogeneous percolation critical value. For $s\leq 2$, it is natural to conjecture that there is a new critical value, $p_c^s(\mathbb{Z}^d)$ for $p_1$, strictly between $p_c(\mathbb{Z}^d)$ and $p_c(\mathbb{Z}^s)$ ; we prove this for large $d$ and $2 \leq s \leq d-3$. For $s=1$, it is natural to conjecture that $p_c^1(\mathbb{Z}^d) =1$, as shown for $d =2$ by Zhang; we prove this for large $d$. Related results for the contact process are also presented.