  The spherical ensemble is a well-known ensemble of N repulsive points on the
two-dimensional sphere, which can realized in various ways (as a random matrix
ensemble, a determinantal point process, a Coulomb gas, a Quantum Hall
state...). Here we show that the spherical ensemble enjoys nearly optimal
convergence properties from the point of view of numerical integration. More
precisely, it is shown that the numerical integration rule corresponding to N
nodes on the two-dimensional sphere sampled in the spherical ensemble is, with
overwhelming probability, nearly a quasi-Monte-Carlo design in the sense of
Brauchart-Saff-Sloan-Womersley (for any smoothness parameter s less than or
equal to two). The key ingredient is a new explicit concentration of measure
inequality for the spherical ensemble.
