The length of the shortest edge of a graph on a sphere
Let \(S^{d}\) denote a unit sphere in the \((d+1)\)-dimensional Euclidean space \(\mathbb{R}^{d+1}\) \((d\geq 1)\). For a simple graph \(G_{E}\) with edge set \(E\), take independent random points \(x_{k}\) \((k\in V(G_{E}))\) on \(S^{d}\), and let \(D_{E}\) be the minimum value of the spherical distance between \(x_{i},x_{j}\) for \(\{i,j\}\in E\). Then \(D_{E}\) is a random variable. The main result of this paper is that \(|E|D_{E}^{d}\) is asymptotically distributed (as \(|E|\rightarrow \infty)\) according to the exponential distribution with mean \(dB(\frac{1}{2},\frac{d}{2})\), where \(B(p,q)\) is the beta function.
scientific article; zbMATH DE number 1884157