Approximation to uniform distribution in $$\mathrm {SO}(3)$$

Abstract

Using the theory of determinantal point processes we give upper bounds for the Green and Riesz energies for the rotation group SO(3), with Riesz parameter up to 3. The Green function is computed explicitly, and a lower bound for the Green energy is established, enabling comparison of uniform point constructions on SO(3). The variance of rotation matrices sampled by a certain determinantal point process is estimated, and formulas for the L 2 -norm of Gegenbauer polynomials with index 2 are deduced, which might be of independent interest.

Weighted $L^2$-Norms of Gegenbauer polynomials

“…Here f is some potential depending only on the distance of two points. In many cases these computations lead to integrals of the form (1) (see [1,3,4,6]).…”

Riesz energy, L2$L^2$ discrepancy, and optimal transport of determinantal point processes on the sphere and the flat torus

“…It was first introduced by Beltrán, Marzo, and Ortega‐Cerdà on the sphere $$\mathbb {S}^d$$ in [14], where they established the precise asymptotics of its expected singular Riesz and logarithmic energies, and studied its separation distance and linear statistics. The energy results have since been extended to projective spaces [5] and to $$\mathrm{SO}(3)$$ [13], where the Green energy was also found. The harmonic ensemble has also been considered on the flat torus in [39], where its expected periodic Riesz energy for $$0<s<d$$ was investigated.…”

On Gegenbauer Point Processes on the Unit Interval

“…The second property implies that the random points exhibit some repulsion, and has been used to give upper bounds on the minimum value of the energy E log (K, n) (and other energies) for different sets: in [2] for the 2-sphere S 2 , in [7] for the d-sphere S d , in [5] (see also [8]) for the complex projective space, in [3] for 2-point homogeneous spaces, in [18] for the flat torus and in [6] for the rotation group SO (3).…”