Approximation on slabs and uniqueness for Bernoulli percolation with a sublattice of defects

Let Ld = (Zd, Ed) be the d-dimensional hypercubic lattice. We consider a model of inhomogeneous Bernoulli percolation on Ld in which every edge inside the s-dimensional sublattice Zs x {0}d-s, 2 < s < d, is open with probability q and every other edge is open with probability p. We prove the uniqueness of the infinite cluster in the supercritical regime whenever p =6 pc(d) and 2 < s < d - 1, full uniqueness when s = d - 1 and that the critical point (p, qc(p)) can be approximated on the phase space by the critical points of slabs, for any p < pc(d), where pc(d) denotes the threshold for homogeneous percolation.