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 less th an or equal to s less than or equal to d - 1) have density p(1) and al l other bonds have fixed density, p(c)(Z(d)), the homogeneous percolat ion critical value. For s greater than or equal to 2, it is natural to conjecture that there is a new critical value, p(c)(s)(Z(d)), for p(1 ), strictly between p(c)(Z(d)) and p(c)(Z(s)); we prove this for large d and 2 less than or equal to s less than or equal to d - 3. For s = 1, it is natural to conjecture that p(c)(1)(Z(d)) = 1, as shown for d = 2 by Zhang; we prove this for large d. Related results for the conta ct process are also presented.