A point process is R-dependent if it behaves independently beyond the minimum distance R. In this paper we investigate uniform positive lower bounds on the avoidance functions of R-dependent simple point processes with a common intensity. Intensities with such bounds are characterised by the existence of Shearer’s point process, the unique R-dependent and R-hard-core point process with a given intensity. We also present several extensions of the Lovász local lemma, a sufficient condition on the intensity andR to guarantee the existence of Shearer’s point process and exponential lower bounds. Shearer’s point process shares a combinatorial structure with the hard-sphere model with radius R, the unique R-hard-core Markov point process. Bounds from the Lovász local lemma convert into lower bounds on the radius of convergence of a high-temperature cluster expansion of the hard-sphere model. This recovers a classic result of Ruelle (1969) on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive approach of Dobrushin (1996).

Point processes (msc 60G55), None of the above, but in MSC2010 section 05Dxx (msc 05D99), Random fields (msc 60G60), Classical equilibrium statistical mechanics (general) (msc 82B05)
Advances in Applied Probability

Hofer-Temmel, C. (2017). Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma. Advances in Applied Probability, 49(1), 1–23.