The Projective Ensemble and Distribution of Points in Odd-Dimensional Spheres

Abstract

We define a determinantal point process on the complex projective space that reduces to the so-called spherical ensemble for complex dimension 1 under identification of the 2-sphere with the Riemann sphere. Through this determinantal point process we propose a point processs in odddimensional spheres that produces fairly well-distributed points, in the sense that the expected value of the Riesz 2-energy for these collections of points is smaller than all previously known bounds.

Approximation to uniform distribution in SO(3)

“…The theory of those processes has been developed in [8]; there one also finds a pseudo-code which samples points based on the dpp -which seems hard to implement. A main feature of the underlying points is that they tend to "repel" each other, and hence have become the theoretical basis of construction of well-distributed points on various symmetric spaces, see for instance [2,6,7,22].…”

“…We point the reader to the excellent monograph [8] for an introduction to point processes, and we briefly summarize part of this material below. As in [7] and [6], we will use only a fraction of the theory.…”

On $p$-frame potentials of determinantal point processes on the sphere

“…As a byproduct, we can see numerically or directly (see Remarks 6 and 9) that the expectation of p-frame potentials of such DPPs {x i } N i=1 is asymptotically smaller than that of Poisson point processes on the sphere, that is, it holds that E FP p,N ({x i } N i=1 ) = N 2 PFP(σ d , p) + o(N ) (N → ∞). Furthermore, although this study focused only on three typical DPPs, recently, Beltrán and Etayo [16,17] proposed various point processes on the sphere, which we will explore in future work.…”

Finite frames, frame potentials and determinantal point processes on the sphere

“…The p-frame potentials of DPPs will be studied in future. Although only three typical DPPs are addressed herein, various point processes on the sphere were recently proposed by Beltrán and Etayo [3,4,5], which will be addressed in future.…”