On the pair correlation statistics for determinantal point processes on the sphere

TL;DR

This paper analyzes the pair correlation statistics of determinantal point processes on the sphere, highlighting their small-scale repulsion and similarity to i.i.d. cases at larger scales.

Contribution

It provides a detailed comparison of pair correlation statistics for determinantal point processes with i.i.d. sampling on the sphere.

Findings

Determinantal point processes exhibit small-scale repulsion.

At larger scales, their statistics align with i.i.d. cases.

Results include analysis of spherical and harmonic ensembles, and jittered sampling.

Abstract

In this paper, we study the expected value of the pair correlation statistics of randomized point configurations on the sphere, with the emphasis on point configurations generated by determinantal point processes. We study the cases of the spherical ensemble, the harmonic ensemble, and jittered sampling, and compare our results with those for the ''truly random'' (i.i.d.) case. Our results give evidence of the small-scale repulsion phenomenon which is characteristic for determinantal point processes, while on larger scales there is good agreement between all our studied cases and the i.i.d. case.