Uniqueness of the infinite component in a random graph with applications to percolation and spin glasses

We extend the theorem of Burton and Keane on uniqueness of the infinite component in dependent percolation to cover random graphs on Z(d) or Z(d) x N with long-range edges. We also study a short-range percolation model related to nearest-neighbor spin glasses on Z(d) or on a slab Z(d) x {0,..., K} and prove both that percolation occurs and that the infinite component is unique for V = Z2 x {0,1} or larger