Georgia Institute of Technology College of Sciences

We study the asymptotic behaviors of the pairwise angles among n randomly and uniformly distributed unit vectors in p-dimensional spaces as the number of points n goes to infinity, while the dimension p is either fixed or growing with n. 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 connections with some open problems in physics and mathematics are also discussed. This is a joint work with Tony Cai and Jianqing Fan.