Lasse Leskelä: Hard-core thinnings of germ-grain models with power-law grain sizes

Random sets with long-range dependence can be generated using a Boolean model with power-law grain sizes. This talk focuses on thinnings of such Boolean models which have the hard-core property that no grains overlap in the resulting germ-grain model. A fundamental question is whether long-range dependence is preserved under such thinnings. To answer this question we study four Matérn-type thinnings of a Poisson germ-grain model where the grains are spheres with a regularly varying size distribution. It turns out that a thinning which favors large grains preserves the slow correlation decay of the original model, whereas a thinning which favors small grains does not. The most interesting finding concerns the case where only disjoint grains are retained, which corresponds to the Matérn type I thinning. In the resulting germ-grain model, typical grains have exponentially small sizes, but rather surprisingly, the long-range dependence property is still present. As a byproduct, we obtain new mechanisms for generating homogeneous and isotropic random point configurations having a power-law correlation decay.

This talk is based on a paper with the same title ( http://arxiv.org/abs/1204.1208 ), joint work with Mikko Kuronen (University of Jyväskylä).