Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma
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).
|MSC||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)|
|Journal||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.