Random Geometric Graph Diameter in the Unit Ball

The unit ball random geometric graph $G=G^d_p(\lambda,n)$ has as its vertices n points distributed independently and uniformly in the unit ball in ${\Bbb R}^d$, with two vertices adjacent if and only if their ℓp-distance is at most λ. Like its cousin the Erdos-Renyi random graph, G has a connectivity threshold: an asymptotic value for λ in terms of n, above which G is connected and below which G is disconnected. In the connected zone we determine upper and lower bounds for the graph diameter of G. Specifically, almost always, ${\rm diam}_p({\bf B})(1-o(1))/\lambda\leq {\rm diam}(G) \leq {\rm diam}_p({\bf B})(1+O((\ln \ln n/{\rm ln}\,n)^{1/d}))/\lambda$, where ${\rm diam}_p({\bf B})$ is the ℓp-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.