A threshold for the size of random caps to cover a sphere
Let the centers of N random spherical caps, each of area \(4\pi\) p(N), be independently and uniformly distributed over the unit sphere. This paper considers the asymptotic behavior, as N tends to infinity, of the probability of completely covering the sphere. The author shows that \(p_ 0(N)=(\log N)/N\) is a threshold function in the sense that: (i) if p(N)\(\leq c p_ 0(N)\) for some constant \(c<1\), then the probability that the sphere is completely covered tends to zero; (ii) if p(N)\(\geq c p_ 0(N)\) for some constant \(c>1\), then for each positive integer n, the probability that every point on the sphere is covered by more than n caps tends to one.
scientific article; zbMATH DE number 4115623