The spherical ensemble is a well-studied determinantal process with a fixed number of points on S 2 . 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).for all continuous, compactly supported functions F : C k → C, where dµ(z) := n π(1+|z| 2 ) n+1 dz and dz denotes the Lebesgue measure on the complex plane C.Krishnapur [22] showed that this random point process is a determinantal point process on complex plane with kernel K (n) (z, w) = (1 + zw) n−1 with respect to the background measure dµ(z), i.e. we havefor every k ≥ 1 and z 1 , . . . , z k ∈ C. We note here that a random point process is said to be a determinantal point process if its k-point correlation functions have determinantal form similar to (1.2). The corresponding kernel K (n) (z, w) is called a correlation kernel of the determinantal point process. We refer to [20] or [21] and references therein for more information on deteminantal point processes. Let S 2 = {p ∈ R 3 : |p| = 1} be the unit two-dimensional sphere centred at the origin in three-dimensional Euclidean space R 3 . Also we let ν denotes the Lebesgue surface area measure on this sphere with total measure 4π. As mentioned in [21], these eigenvalues are best thought of as points on S 2 , using stereographic projection. Let g be the stereographic projection of the sphere S 2 from the north pole onto the plane {(t 1 , t 2 , 0); t 1 , t 2 ∈ R}. If we let P i = g −1 (λ i ) for 1 ≤ i ≤ n then the vector (P 1 , . . . , P n ), in uniform random order, has the joint density Const.i

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We prove that every strongly Rayleigh sequence of random variables has a trivial tail σ-field. This is a generalization of an earlier result of Lyons which established tail triviality for discrete determinantal processes. Unlike Lyons' proof, our approach is probabilistic in nature and relies on negative dependence properties of strongly Rayleigh processes.