  We study the spherical cap packing problem with a probabilistic approach.
Such probabilistic considerations result in an asymptotic sharp universal
uniform bound on the maximal inner product between any set of unit vectors and
a stochastically independent uniformly distributed unit vector. When the set of
unit vectors are themselves independently uniformly distributed, we further
develop the extreme value distribution limit of the maximal inner product,
which characterizes its uncertainty around the bound. As applications of the
above asymptotic results, we derive (1) an asymptotic sharp universal uniform
bound on the maximal spurious correlation, as well as its uniform convergence
in distribution when the explanatory variables are independently Gaussian
distributed; and (2) an asymptotic sharp universal bound on the maximum norm of
a low-rank elliptically distributed vector, as well as related limiting
distributions. With these results, we develop a fast detection method for a
low-rank structure in high-dimensional Gaussian data without using the spectrum
information.
