On Maximal Hard-Core Thinnings of Stationary Particle Processes

Abstract

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.

Limit theory of combinatorial optimization for random geometric graphs

“…A related quantity is the maximum possible intensity of a hard-core stationary thinning of the whole of H λ ; we denote this quantity here by ρ(λ). See [24] for further details on hard-core stationary thinnings. It is not hard to see that ρ(λ) ≤ λρ(λ), and we conjecture that in fact equality holds here.…”