Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma

Abstract

A point process isR-dependent if it behaves independently beyond the minimum distanceR. In this paper we investigate uniform positive lower bounds on the avoidance functions ofR-dependent simple point processes with a common intensity. Intensities with such bounds are characterised by the existence of Shearer's point process, the uniqueR-dependent andR-hard-core point process with a given intensity. We also present several extensions of the Lovász local lemma, a sufficient condition on the intensity andRto 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 radiusR, the uniqueR-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).

Disagreement percolation for the hard-sphere model

“…In one dimension, disagreement percolation (9a) replicates Tonks' classic result of the complete absence of phase transitions via virial expansion methods [3,12,15,26]. In terms of the activity, it is known that the radius of the cluster expansion is exactly [3,11,15]…”