On random fields of segments and random mosaics on the plane

We consider random fields of segments on the plane and random mosaics (i.e. such random fields of segments that, with probability 1, partition the plane into convex bounded polygons). The random fields under consideration are assumed to be homogeneousand isotropic, i.e. the probability measure is invariant relative to Euclidean transformations of the plane.
The main objects of the investigation are “stars” — collections of segments forming random fields which have a common point.