从对称破缺到二维Voronoi曲面的Poisson点过程：六边形的类属性质
From symmetry break to Poisson point process in 2D Voronoi tessellations: the generic nature of hexagons
摘要：
我们的桥梁的定期广场和蜂窝Voronoi镶嵌的平面的Poisson-Voronoi的情况下的属性，从而分析在一个共同的框架内的破坏过程和均匀随机分布的镶嵌生成点的方法。我们认为合奏模拟产生的点，其经常的位置被扰动通过高斯噪声控制的参数α的镶嵌。我们研究了Voronoi细胞的边数、面积和周长。对于α>0，六边形是最常见的细胞类别，并且2参数伽马分布很好地描述了几何特征的统计。由于噪声导致的对称性破坏破坏了正方形镶嵌，而蜂窝六边形镶嵌非常稳定，并且对于α <0.1的小但有限的噪声，所有Voronoi单元都是六边形。对于中等量的高斯噪声，特定的未扰动的细分的存储器丢失，因为两个扰动的细分的统计是不可区分的。当alpha>2时，结果收敛于Poisson-Voronoi细分的结果。n边单元的性质随α而变化，直到α>2时达到泊松-沃罗诺伊极限。Desch法周长被证实是无效的，并建立了平方根依赖于n。限制到六边形单元格的单元格面积和周长的系综平均值与完整系综平均值相一致;这可能意味着边的数量作为一个热力学状态变量在n=6左右波动;这加强了六边形的想法，超越了它们无处不在的数值突出性，可以被视为2D Voronoi镶嵌中的通用多边形。
英文摘要：
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 consider ensemble simulations of tessellations generated by points whose regular positions are perturbed through a Gaussian noise controlled by the parameter alpha. We study the number of sides, the area, and the perimeter of the Voronoi cells. For alpha>0, hexagons are the most common class of cells, and 2-parameter gamma distributions describe well the statistics of the geometrical characteristics. The symmetry break due to 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. For a moderate amount of Gaussian noise, memory of the specific unperturbed tessellation is lost, because the statistics of the two perturbed tessellations is indistinguishable. When alpha>2, results converge to those of Poisson-Voronoi tessellations. The 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 is established. 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; this reinforces the idea that hexagons, beyond their ubiquitous numerical prominence, can be taken as generic polygons in 2D Voronoi tessellations.
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