The length of the shortest edge of a graph on a sphere

概要

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Let $S^d$ denote a unit sphere in the $(d+1)$-dimensional Euclidean space $\bold R^{d+1} (d\geq 1)$. For a simple graph $G_{\scr E}$ with edge set $\scr E$, take independent random points $x_k, k\in V(G_{\scr E})$, on $S^d$, and let $D_{\scr E}$ be the minimum value of the spherical distance between $x_i,x_j$ for $\{i,j\}\in\scr E$. We prove that $|\scr E|D^d_{\scr E}$ is asymptotically (as $|\scr E|\to\infty$) distributed to the exponential distribution with mean $dB(\frac 12,\frac d2)$, where $B(p,q)$ is the beta function.
Elsevier Science B.V.の論文
2002-08-00