| Research Abstract |
When a configuration of many particles is given in the space, it is the important subject of "Spatial Statistics" to clarify the statistical properties of the particle configuration. The approach in which the statistical distributions of Voronoi cells for a given particle configuration are studied is one of the effective methods in this area. It is known that a natural null hypothesis for a given configuration is a Poisson point processes. Correspondingly, it is indispensable to clarify the statistical properties of Voronoi cells for Poisson point processes (i.e., Poisson Voronoi cells) in our approach. In this research project, we aimed at obtaining the statistical properties of Poisson Voronoi cells accurately as far as possible through computer simulations even in higher dimensional space. During the period of the project, we succeeded to obtain Poisson Voronoi cells up to five-dimensions where the sample size is ten million for d = 2, and five million for d = 3,4,5,respectively. We then made histograms of volume, surface area and other characteristics of Poisson Voronoi cells, fitted a 3-parameter generalized Gamma distribution to them, and obtained experimental expressions for the statistical distributions of these characteristics. We also checked some theoretical moment values against those of our experimental results and found good coincidences. We found some new facts which arise for Poisson Voronoi cells for d = 4 and 5. Parts of our results were published as papers, and presented at international and national meetings.
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