Valerio Lucarini
From symmetry break to Poisson point process in 2D Voronoi tessellations: the generic nature of hexagons
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ABSTRACT. We bridge the properties of the regular square and honeycomb Voronoi tessellations of the plane to those of the Poisson-Voronoi case, thus analyzing in a common framework symmetry-break processes and the approach to uniformly random distributions of tessellation-generating points. We resort to ensemble simulations of tessellations generated by points whose regular positions is perturbed through a Gaussian noise controlled by the parameter alpha. We analyze the number of sides, the area, and the perimeter of the Voronoi cells. For alpha>0, hexagons constitute the most common class of cells, and 2-parameter gamma distributions provide an efficient description of statistical properties of the analyzed geometrical characteristics. The symmetry break induced by the introduction of noise destroys the square tessellation, whereas the honeycomb hexagonal tessellation is very stable and all Voronoi cells are hexagon for small but finite noise with alpha<0.1. Several statistical signatures of the symmetry break are evidenced. For a moderate amount of Gaussian noise (alpha>0.5), memory of the specific initial unperturbed state is lost, because the statistics of the two perturbed regular tessellations is indistinguishable. When alpha>2, results converge to those of Poisson-Voronoi tessellations. The geometrical properties of n-sided cells change with alpha until the Poisson-Voronoi limit is reached for alpha>2. The Desch law for perimeters is confirmed to be not valid and a square root dependence on n, which allows an easy link to the Lewis law for areas, is established. Finally, the ensemble mean of the cells area and perimeter restricted to the hexagonal cells coincides with the full ensemble mean; this might imply that the number of sides acts as a thermodynamic state variable fluctuating about n=6, and this reinforces the idea that hexagons, beyond their ubiquitous numerical prominence, can be taken as generic polygons in 2D Voronoi tessellations.