Planar discrete birth-growth Poisson–Voronoi tessellations with the von Neumann neighbourhood

Abstract

Poisson-Voronoi tessellations are widely used as the generic model for studying various birth-growth processes and resulting morphologies in physics, chemistry, materials science, and related fields. This paper studies planar discrete Poisson-Voronoi tessellations constructed directly by the growth to impingement of random square germs. They materially dier from similar tessellations constructed of the nearest tile loci according to the basic definition. The boundary structure is described in detail. Its peculiarities are used to extend the concept of Gabriel edges to the considered discrete case and also to quantify this concept. The averaged percentage of Gabriel edges appears to be practically independent of the germs density, G = 70%. The studied densities range from 0.01 to 0.000 01. Statistical results are presented for the whole tessellation and also for subsets of random domains with the given number of edges ν. Two sets of results are compared: for edges of each random domain arranged from the longest to the shortest and for edges arranged from the nearest to the most distant. Averaged distances to neighbours in the metric determined by the growth mode of islands are compared with that in the Euclidean metric. Also, the cyclic sequences of edge lengths of random domains are examined. The linearity with respect to ν is revealed for four scaling-related characteristics: the area of random domains, the perimeter length of random domains, the area of complete concentric belts, and the coordinates of maxima of kinetic curves.