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Random sets with long-range dependence can be generated using a Boolean modelwith power-law grain sizes. We study thinnings of such Boolean models whichhave the hard-core property that no grains overlap in the resulting germ-grainmodel. A fundamental question is whether long-range dependence is preservedunder such thinnings. To answer this question we study four natural thinningsof a Poisson germ-grain model where the grains are spheres with a regularlyvarying size distribution. We show that a thinning which favors large grainspreserves the slow correlation decay of the original model, whereas a thinningwhich favors small grains does not. Our most interesting finding concerns thecase where only disjoint grains are retained, which corresponds to thewell-known Mat\'ern type I thinning. In the resulting germ-grain model, typicalgrains have exponentially small sizes, but rather surprisingly, the long-rangedependence property is still present. As a byproduct, we obtain new mechanismsfor generating homogeneous and isotropic random point configurations having apower-law correlation decay.

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