2017-03-17
Shearer's point process, the hard-sphere model, and a continuum Lovász local lemma
Publication
Publication
Advances in Applied Probability , Volume 49 - Issue 1 p. 1- 23
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).
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| doi.org/10.1017/apr.2016.76 | |
| Advances in Applied Probability | |
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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. doi:10.1017/apr.2016.76 |
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