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 ofR ². An expression for the probabilities π,k=0, 1, 2,... to havek hits on an interval of lengtht 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, hereP _i is the point process of intersections of edges of the mosaic with a fixed line of direction α and the markψ _i is the intersection angle atP _i . The method is based on the results of combinatorial integral geometry.