Aggregate and Fractal Tessellations
Résumé
Consider a sequence of stationary tessellations {Theta^n, n=0,1,...} of R^d consisting of cells {C^n(x_i^n)} with the nuclei {x_i^n}. An aggregate cell of level one, C_0^1(x_i^0), is the result of merging the cells of Theta^1 whose nuclei lie in C^0(x_i^0). An aggregate tessellation Theta_0^n consists of the aggregate cells of level n, C_0^n(x_i^0), defined recursively by merging those cells of Theta^n whose nuclei lie in C_0^n-1(x_i^0). We find an expression for the probability for a point to belong to a typical aggregate cell and obtain bounds for the probability of cell's expansion and extinction. We give necessary conditions for the limit tessellation to exist as n to infinity and provide upper bounds for the Hausdorff dimension of its fractal boundary and for the spherical contact distribution function in the case of Poisson-Voronoi tessellations {Theta^n}.