Extreme Values and Convergence of the Voronoi Entropy for 2D Random Point Processes and for Long-Range Order
Mark Frenkel, Irina Legchenkova, Edward Bormashenko, Shraga Shoval, Michael Nosonovsky
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.
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…
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Taxonomy
TopicsPoint processes and geometric inequalities · Random Matrices and Applications · Statistical Mechanics and Entropy