Percolation and contact processes with low-dimensional inhomogeneity

We consider inhomogeneous nearest neighbor Bernoulli bond percolation on ℤ^d where the bonds in a fixed s-dimensional hyperplane (1 ≤ s ≤ d - 1) have density p₁ and all other bonds have fixed density, p_c(ℤ^d), the homogeneous percolation critical value. For s ≥ 2, it is natural to conjecture that there is a new critical value, p^s_c(ℤ^d), for p₁, strictly between p_c(ℤ^d) and p_c(ℤ^s); we prove this for large d and 2 ≤ s ≤ d - 3. For s = 1, it is natural to conjecture that p¹_c(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.