# Hyperuniformity on spherical surfaces

## TL;DR

This paper investigates hyperuniform point distributions on spherical surfaces, analyzing local particle number variance to characterize order and disorder, extending hyperuniformity concepts from Euclidean to curved geometries.

## Contribution

It introduces a method to characterize hyperuniformity on spherical surfaces using local number variance scaling, addressing a gap in existing literature.

## Key findings

- Hyperuniformity can be characterized on spherical surfaces via number variance scaling.
- Disordered hyperuniform systems like avian retina photoreceptors are analyzed.
- The approach applies to regular, uniform, and equilibrium particle configurations.

## Abstract

In this work we present a study on the characterization of ordered and disordered hyperuniform point distributions on spherical surfaces. In spite of the extensive literature on disordered hyperuniform systems in Euclidean geometries, to date few works have dealt with the problem of hyperuniformity in curved spaces. As a matter of fact, some systems that display disordered hyperuniformity, like the space distribution of photoreceptors in avian retina, actually occur on curved surfaces. Here we will focus on the local particle number variance and its dependence on the size of the sampling window (which we take to be a spherical cap) for regular and uniform point distributions, as well as for equilibrium configurations of fluid particles interacting through Lennard-Jones, dipole-dipole and charge-charge potentials. We will show how the scaling of the local number variance enables the characterization of hyperuniform point patterns also on spherical surfaces.