On the distance to the giant component along a straight line in a two-dimensional percolation model

The supercritical regime of a percolation model refers to the range of probabilities (discrete) or densities (continuous) above a critical value for which there exists a unique unbounded cluster almost surely. In this paper, we provide an upper bound to the linear distance from the origin to this giant connected component for both the discrete and the continuous (Boolean) model in two-dimensions. By modeling a dense wireless sensor network with a supercritical Boolean model, our result bounds the distance traveled by a target moving in a straight line before it is detected by a node who can relay the alert through a multihop path to the sink. This result incorporates a solidified definition of detection requiring that the intrusion alert successfully reach the central authority.