We consider inhomogeneous nearest neighbor Bernoulli bond percolation on ℤd where the bonds in a fixed s-dimensional hyperplane (1 ≤ s ≤ d - 1) have density p1 and all other bonds have fixed density, pc(ℤd), the homogeneous percolation critical value. For s ≥ 2, it is natural to conjecture that there is a new critical value, psc(ℤd), for p1, strictly between pc(ℤd) and pc(ℤs); we prove this for large d and 2 ≤ s ≤ d - 3. For s = 1, it is natural to conjecture that p1c(ℤd) = 1, as shown for d = 2 by Zhang; we prove this for large d. Related results for the contact process are also presented.
All Science Journal Classification (ASJC) codes
- Statistics and Probability
- Statistics, Probability and Uncertainty