Riesz and Green energy on projective spaces

Abstract

In this paper we study Riesz, Green and logarithmic energy on two-point homogeneous spaces. More precisely we consider the real, the complex, the quaternionic and the Cayley projective spaces. For each of these spaces we provide upper estimates for the mentioned energies using determinantal point processes. Moreover, we determine lower bounds for these energies of the same order of magnitude.

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

“…The 𝐿 2 discrepancy on ℂℙ 𝑑 was studied in [43,45], and the Riesz energy in [5,11,22]. The optimal order of the former for any set of 𝑁 points is [43, Theorem 2.2]…”

“…It was first introduced by Beltrán, Marzo, and Ortega-Cerdà on the sphere 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 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.…”

“…The proof of all our results rely on the explicit formulae for the Laplace eigenfunctions on 𝕊 𝑑 and 𝕋 𝑑 . We mention without further details that similarly to [5], our approach could be extended to other two-point homogeneous spaces, on which the Laplace eigenfunctions are also explicitly known. We only comment on the complex projective space ℂℙ 𝑑 , in particular on the expected value of the 𝐿 2 discrepancy of the projective ensemble (Corollary 3).…”