  The spherical ensemble is a well-studied determinantal process with a fixed
number of points on the sphere. The points of this process correspond to the
generalized eigenvalues of two appropriately chosen random matrices, mapped to
the surface of the sphere by stereographic projection. This model can be
considered as a spherical analogue for other random matrix models on the unit
circle and complex plane such as the circular unitary ensemble or the Ginibre
ensemble, and is one of the most natural constructions of a (statistically)
rotation invariant point process with repelling property on the sphere. In this
paper we study the spherical ensemble and its local repelling property by
investigating the minimum spacing between the points and the area of the
largest empty cap. Moreover, we consider this process as a way of distributing
points uniformly on the sphere. To this aim, we study two "metrics" to measure
the uniformity of an arrangement of points on the sphere. For each of these
metrics (discrepancy and Riesz energies) we obtain some bounds and investigate
the asymptotic behavior when the number of points tends to infinity. It is
remarkable that though the model is random, because of the repelling property
of the points, the behavior can be proved to be as good as the best known
constructions (for discrepancy) or even better than the best known
constructions (for Riesz energies).
