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Journal of Statistical Physics

, Volume 118, Issue 5–6, pp 1151–1261 | Cite as

The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lovász Local Lemma

  • Alexander D. Scott
  • Alan D. Sokal
Article

Abstract

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovász local lemma in probabilistic combinatorics. We show that the conclusion of the Lovász local lemma holds for dependency graph G and probabilities {p x } if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p x }. Furthermore, we show that the usual proof of the Lovász local lemma – which provides a sufficient condition for this to occur – corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer(98) and explicitly by Dobrushin.(37,38) We also present some refinements and extensions of both arguments, including a generalization of the Lovász local lemma that allows for ‘‘soft’’ dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.

Keywords

Graph lattice gas hard-core interaction independent-set polynomial polymer expansion cluster expansion Mayer expansion Lovász local lemma probabilistic method 

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© Springer Science+Business Media, Inc. 2005

Authors and Affiliations

  1. 1.Department of MathematicsUniversity College LondonLondonEngland
  2. 2.Department of PhysicsNew York UniversityNew YorkUSA

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