# The projective ensemble and distribution of points in odd-dimensional spheres

## TL;DR

This paper introduces a determinantal point process on complex projective spaces that generates well-distributed points in odd-dimensional spheres, achieving lower expected Riesz 2-energy than previous methods.

## Contribution

It defines a new determinantal point process on complex projective space that extends the spherical ensemble to odd-dimensional spheres, improving point distribution quality.

## Key findings

- Expected Riesz 2-energy is lower than all previously known bounds.
- The process produces well-distributed points in odd-dimensional spheres.
- The approach generalizes the spherical ensemble to higher dimensions.

## 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 odd-dimensional 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.