A threshold for the size of random caps to cover a sphere
type: Item
instance of: scholarly article
title: A threshold for the size of random caps to cover a sphere
zbMATH Open document ID: 0681.60017
DOI: 10.1007/BF00049424
author: H. Maehara
published in: Annals of the Institute of Statistical Mathematics
publication date: 1988-01-01T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 60D05
zbMATH DE Number: 4115623
zbMATH Keywords: random spherical caps
zbMATH Keywords: asymptotic behavior
reviewed by: William J. Firey
MaRDI profile type: Publication
cites work: Q3291025
cites work: On the number of induced subgraphs of a random graph
cites work: Q4110292
cites work: Q4160140
description: scientific article; zbMATH DE number 4115623