Connectivity for line-of-sight networks in higher dimensions

Let T be a d-dimensional toroidal grid of n points. For a given range parameter ω, 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 ω apart, where distance is measured using the `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.