Combinatorial Decompositions and Homogeneous Geometrical Processes

Abstract

This paper considers line processes and random mosaics. The processes are assumed invariant with respect to the group of translations of R ². An expression for the probabilities π_k(t, α), k = 0,1,2,… to have k hits on an interval of length t taken on a ‘typical line of direction α’ (the hits are produced by other lines of the process) is obtained. Also, the distribution of a length of a ‘typical edge having direction α’ in terms of the process {P _i, ψ_i} is found, here P _i is the point process of intersections of edges of the mosaic with a fixed line of direction α and the mark ψ_i is the intersection angle at P _i. The method is based on the results of combinatorial integral geometry.