On the probability estimates of quadrature rules from uniformly sampled points on spheres

In this paper, we provide detailed discussions on the probability estimates of quadrature rules from uniformly sampled points on spheres. Besides gathering relevant lemmas in the literature to derive probability estimates on the existence of exact quadrature rules for spherical harmonics, we provide additional estimates with finer characterizations based on probabilistic quantities related to the measure and the diameter of Voronoi cells. Specifically, our estimates provide additional affirmative answers to certain relations between the number of sampled points and the degree of spherical harmonics. We further investigate the problem of setting the number of points to be of order t^d. Simple analysis based on our estimates suggests that the constant in the order cannot be fixed for all t and should increase as t increases. This is empirically verified in our experiments. © 2026 World Scientific Publishing Company.