A random geometric graph, @math , is formed by choosing @math points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most @math . For a given constant @math , we show that @math is a distance threshold function for @math to have a connected subgraph on @math points. Based on this, we show that @math is a distance threshold for @math to be plane, and @math is a distance threshold to be planar. We also investigate distance thresholds for @math to have a non-crossing edge, a clique of a given size, and an independent set of a given size.
We look at a model of random graphs suggested by Gilbert: given an integer @math and @math , scatter @math vertices independently and uniformly on a metric space, and then add edges connecting pairs of vertices of distance less than @math apart.We consider the asymptotics when the metric space is the interval [0, 1], and @math is a function of @math , for @math . We prove that every upwards closed property of (ordered) graphs has at least a weak threshold in this model on this metric space. (But we do find a metric space on which some upwards closed properties do not even have weak thresholds in this model.) We also prove that every upwards closed property with a threshold much above connectivity's threshold has a strong threshold. (But we also find a sequence of upwards closed properties with lower thresholds that are strictly weak.)  Random geometric graphs result from taking n uniformly distributed points in the unit cube, [0,1]d, and connecting two points if their Euclidean distance is at most r, for some prescribed r. We show that monotone properties for this class of graphs have sharp thresholds by reducing the problem to bounding the bottleneck matching on two sets of @math points distributed uniformly in [0,1]d. We present upper bounds on the threshold width, and show that our bound is sharp for d = 1 and at most a sublogarithmic factor away for d ≥ 2. Interestingly, the threshold width is much sharper for random geometric graphs than for Bernoulli random graphs. Further, a random geometric graph is shown to be a subgraph, with high probability, of another independently drawn random geometric graph with a slightly larger radius; this property is shown to have no analogue for Bernoulli random graphs.  In this review paper, we shall discuss some recent results concerning several models of random geometric graphs, including the Gilbert disc model G r , the k-nearest neighbour model G k nn and the Voronoi model G P . Many of the results concern finite versions of these models. In passing, we shall mention some of the applications to engineering and biology.  A way of improving the efficiency of gland seals of a large steam turbine comprising injecting into the seals a sacrificial bleed of low-energy fluid. This reduces the energy leakage of the gland seals and thus increases overall power plant efficiency. We give a characterization of vertex-monotone properties with sharp thresholds in a Poisson random geometric graph or hypergraph. As an application we show that a geometric model of random k-SAT exhibits a sharp threshold for satisfiability.