多面体Voronoi胞

Explicit Construction of the Voronoi and Delaunay Cells of W(An) and W(Dn) Lattices and Their Facets

Voronoi and Delaunay (Delone) cells of the root and weight lattices of the Coxeter-Weyl groups W(an) and W(dn) are constructed. The face centered cubic (fcc) and body centered cubic (bcc)lattices are obtained in this context. Basic definitions are introduced such as parallelotope, fundamental simplex, contact polytope, root polytope, Voronoi cell, Delone cell, n-simplex, n-octahedron (cross polytope), n-cube and n-hemicube and their volumes are calculated. Voronoi cell of the root lattice is constructed as the dual of the root polytope which turns out to be the union of Delone cells. It is sho…

A universal law for Voronoi cell volumes in infinitely large maps

We discuss the volume of Voronoi cells defined by two marked vertices picked randomly at a fixed given mutual distance 2s in random planar quadrangulations. We consider the regime where the mutual distance 2s is kept finite while the total volume of the quadrangulation tends to infinity. In this regime, exactly one of the Voronoi cells keeps a finite volume, which scales as s^4 for large s. We analyze the universal probability distribution of this, properly rescaled, finite volume and present an explicit formula for its Laplace transform.

Statistics of the Voronoi cell perimeter in large bi-pointed maps

We study the statistics of the Voronoi cell perimeter in large bi-pointed planar quadrangulations. Such maps have two marked vertices at a fixed given distance $2s$ and their Voronoi cell perimeter is simply the length of the frontier which separates vertices closer to one marked vertex than to the other. We characterize the statistics of this perimeter as a function of $s$ for maps with a large given volume $N$ both in the scaling limit where $s$ scales as $N^{1/4}$, in which case the Voronoi cell perimeter scales as $N^{1/2}$, and in the local limit where $s$ remains finite, in which case th…

Prototiles and Tilings from Voronoi and Delone cells of the Root Lattice A_n

We exploit the fact that two-dimensional facets of the Voronoi and Delone cells of the root lattice A_n in n-dimensional space are the identical rhombuses and equilateral triangles respectively.The prototiles obtained from orthogonal projections of the Voronoi and Delaunay (Delone) cells of the root lattice of the Coxeter-Weyl group W(a)_n are classified. Orthogonal projections lead to various rhombuses and several triangles respectively some of which have been extensively discussed in the literature in different contexts. For example, rhombuses of the Voronoi cell of the root lattice A_4 proj…

On a conjecture by Chapuy about Voronoi cells in large maps

In a recent paper, Chapuy conjectured that, for any positive integer k, the law for the fractions of total area covered by the k Voronoi cells defined by k points picked uniformly at random in the Brownian map of any fixed genus is the same law as that of a uniform k-division of the unit interval. For k=2, i.e. with two points chosen uniformly at random, it means that the law for the ratio of the area of one of the two Voronoi cells by the total area of the map is uniform between 0 and 1. Here, by a direct computation of the desired law, we show that this latter conjecture for k=2 actually hol…

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

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。