# On Maximal Hard-Core Thinnings of Stationary Particle Processes

## Abstract

The present paper studies existence and distributional uniqueness of subclasses of stationary hard-core particle systems arising as thinnings of stationary particle processes. These subclasses are defined by natural maximality criteria. We investigate two specific criteria, one related to the intensity of the hard-core particle process, the other one being a local optimality criterion on the level of realizations. In fact, the criteria are equivalent under suitable moment conditions. We show that stationary hard-core thinnings satisfying such criteria exist and are frequently distributionally unique. More precisely, distributional uniqueness holds in subcritical and barely supercritical regimes of continuum percolation. Additionally, based on the analysis of a specific example, we argue that fluctuations in grain sizes can play an important role for establishing distributional uniqueness at high intensities. Finally, we provide a family of algorithmically constructible approximations whose volume fractions are arbitrarily close to the maximum.

## Keywords

Particle process Hard-core process Stochastic domination Thinning## Mathematics Subject Classification

60G55 60D05## Notes

### Acknowledgements

We thank the anonymous referee for the time and energy invested into producing a very thorough and detailed report. The suggestions and comments made in this report substantially improved the quality of the article. This research publication was funded by LMU Munich’s Institutional Strategy LMUexcellent within the framework of the German Excellence Initiative.

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