Connectivity for line-of-sight networks in higher dimensions

Let T be a d-dimensional toroidal grid of n(d) points. For a given range parameter omega, and a positive integer k <= d, we say that two points in T are mutually visible if they differ in at most k coordinates and are a distance at most omega apart, where distance is measured using the l(p) norm. We obtain a random d-dimensional line-of-sight graph G by placing a node at each point in T independently with some fixed probability p* and connecting all pairs of mutually visible nodes. We prove an asymptotically tight connectivity result for this random graph.