# Extreme Values and Convergence of the Voronoi Entropy for 2D Random Point Processes and for Long-Range Order

## TL;DR

This paper studies how entropy behaves in 2D random and ordered point patterns, revealing how entropy changes with structure and size.

## Contribution

The paper introduces new insights into Voronoi Entropy's behavior in random and hyperuniform point sets, linking it to long-range order.

## Key findings

- The Voronoi Entropy (VE) ranges from S = 0 for ordered sets to S = 1.69 for random sets with n > 100 polygons.
- VE reaches saturation at a normalized radius R = 5.5 with 96 ± 6 points.
- VE values can exceed S = 1.69 for certain seed point patterns, indicating deviations from randomness.

## Abstract

We investigate the asymptotic maximum value and convergence of the Voronoi Entropy (VE) for a 2D random point process (S = 1.690 ± 0.001) and point sets with long-range order characterized by hyperuniformity. We find that for the number of polygons of about n > 100, the VE range is between S = 0 (ordered set of seed points) and S = 1.69 (random set of seed points). For circular regions with the dimensionless radius R normalized by the average distance between points, we identify two limits: Limit-1 (R = 2.5, 16 ± 6 points) is the minimum radius, for which it is possible to construct a Voronoi diagram, and Limit-2 (R = 5.5, 96 ± 6 points) at which the VE reaches the saturation level. We also discuss examples of seed point patterns for which the values of VE exceed the asymptotic value of S > 1.69. While the VE accounts only for neighboring polygons, covering the 2D plane imposes constraints on the number of polygons and the number of edges in polygons. Consequently, unlike the conventional Shannon Entropy, the VE captures some long-range order properties of the system. We calculate the VE for several hyperuniform sets of points and compare it with the values of exponents of collective density variables characterizing long-range correlations in the system. We show that the VE correlates with the latter up to a certain saturation level, after which the value of the VE falls to S = 0, and we explain this phenomenon.