## Hard-core thinnings of germ–grain models with power-law grain sizes

*Advances in Applied Probability*, 45(3):595–625, 2013.- doi:10.1239/aap/1377868531
- arXiv:1204.1208.

### Abstract

Random sets with long-range dependence can be generated using a Boolean model with power-law grain sizes. We study 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 natural thinnings of a Poisson germ–grain model where the grains are spheres with a regularly varying size distribution. We show 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. Our most interesting finding concerns the case where only disjoint grains are retained, which corresponds to the well-known 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.

**Keywords**: regular variation, Boolean model, marked Poisson process, germ–grain model, hard-core
model, hard-sphere model

**AMS subject classification**: 60D05, 60G55