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Shearer’s point process, the hard-sphere model, and a continuum Lovász local lemma. (English) Zbl 1425.60049
Summary: 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 and $$R$$ 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 D. Ruelle [Statistical mechanics. Rigorous results. New York etc.: W. A. Benjamin (1969; Zbl 0177.57301)] on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive approach of R. L. Dobrushin [Transl., Ser. 2, Am. Math. Soc. 177, 59–81 (1996; Zbl 0873.60074)].
##### MSC:
 60G55 Point processes (e.g., Poisson, Cox, Hawkes processes) 60G60 Random fields 82B05 Classical equilibrium statistical mechanics (general) 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
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