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

TL;DR

This paper investigates how different hard-core thinning methods affect long-range dependence in germ-grain models with power-law grain sizes, revealing that some thinnings preserve the slow decay of correlations.

Contribution

It demonstrates that certain thinnings, especially Matérn type I, maintain long-range dependence despite size restrictions, providing new insights into generating correlated random structures.

Findings

Large-grain favoring thinning preserves correlation decay

Small-grain favoring thinning does not preserve long-range dependence

Matérn type I thinning retains long-range dependence despite small typical grain sizes

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 Mat\'ern 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.