On the size distribution of Poisson Voronoi cells

• Published in: Physica A Vol. 385; no. 2; pp. 518 - 526
• Main Authors: Ferenc, Járai-Szabó, Néda, Zoltán
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 15.11.2007
• Subjects: Cell-size distribution; Monte Carlo methods; Voronoi diagrams; 02.50.Ng; Monte Carlo methods; Cell-size distribution; Voronoi diagrams; 02.10.−v; 89.75.Kd
• ISSN: 0378-4371; 1873-2119
• DOI: 10.1016/j.physa.2007.07.063

Poisson Voronoi diagrams are useful for modeling and describing various natural patterns and for generating random lattices. Although this particular space tessellation is intensively studied by mathematicians, in two- and three-dimensional (3D) spaces there is no exact result known for the size distribution of Voronoi cells. Motivated by the simple form of the distribution function in the 1D case, a simple and compact analytical formula is proposed for approximating the Voronoi cell's size-distribution function in the practically important 2D and 3D cases as well. Denoting the dimensionality of the space by d ( d = 1 , 2 , 3 ) the f ( y ) = Const * y ( 3 d - 1 ) / 2 exp ( - ( 3 d + 1 ) y / 2 ) compact form is suggested for the normalized cell-size distribution function. By using large-scale computer simulations the viability of the proposed distribution function is studied and critically discussed.