title: Distributions of Angles in Random Packing on Spheres.
abstract: This paper studies the asymptotic behaviors of the pairwise angles amongnrandomly and uniformly distributed unit vectors in [Formula: see text] as the number of pointsn→ ∞, while the dimensionpis either fixed or growing withn. For both settings, we derive the limiting empirical distribution of the random angles and the limiting distributions of the extreme angles. The results reveal interesting differences in the two settings and provide a precise characterization of the folklore that "all high-dimensional random vectors are almost always nearly orthogonal to each other". Applications to statistics and machine learning and connections with some open problems in physics and mathematics are also discussed.
authors.affiliation: Statistics Department The Wharton School University of Pennsylvania Philadelphia, PA 19104, USA TCAI@WHARTON.UPENN.EDU.
authors.forename: Tony
authors.initials: T
authors.lastname: Cai
authors.affiliation: Department of Operation Research and Financial Engineering Princeton University Princeton, NJ 08540, USA JQFAN@PRINCETON.EDU.
authors.forename: Jianqing
authors.initials: J
authors.lastname: Fan
authors.affiliation: School of Statistics University of Minnesota Minneapolis, MN 55455, USA TJIANG@STAT.UMN.EDU.
authors.forename: Tiefeng
authors.initials: T
authors.lastname: Jiang
keywords: empirical law; extreme-value distribution; maximum of random variables; minimum of random variables; packing on sphere; random angle; uniform distribution on sphere