The Length of the Shortest Edge of a Graph on a Sphere

Let Sddenote a unit sphere in the (d + 1)-dimensional Euclidean spaceRd + 1 (d ≥ 1). For a simple graphGE with edge set E, take independent random points xk,k ∈ V(GE), on Sd, and letDE be the minimum value of the spherical distance between xi,xj for { i, j } ∈ E. We prove that | E |DEd is asymptotically (as | E | → ∞ ) distributed according to the exponential distribution with mean dB(1 / 2,d / 2), where B(p, q) is the beta function.