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

Part of:Equilibrium statistical mechanicsStochastic processesGraph theory

Published online by Cambridge University Press:  17 March 2017

Christoph Hofer-Temmel
Christoph Hofer-Temmel*
Affiliation:
VU University Amsterdam
*
* Current address: , c/o FMW, MPC 10A, Postbus 10000, 1780 CA Den Helder, The Netherlands. Email address: math@temmel.me
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Abstract

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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 Ruelle (1969) on the uniqueness of the Gibbs measure of the hard-sphere model via an inductive approach of Dobrushin (1996).

Keywords

Avoidance functionthe Lovász local lemmahard-sphere modelpartition functionR-dependent point processR-hard-core

MSC classification

Primary:60G55: Point processes
Secondary:60G60: Random fields82B05: Classical equilibrium statistical mechanics (general)05C69: Dominating sets, independent sets, cliques

Information

Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 1 , March 2017, pp. 1 - 23
Copyright
Copyright © Applied Probability Trust 2017 

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