2005
DOI: 10.1007/s10955-004-2055-4
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The Repulsive Lattice Gas, the Independent-Set Polynomial, and the Lov�sz Local Lemma

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… Show more

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Cited by 174 publications
(226 citation statements)
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References 88 publications
(128 reference statements)
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“…There is a deep connection between the Lovász Local Lemma [5] and the partition function of the abstract polymer system at negative real fugacity discovered by Scott & Sokal [13]. As already done for Fernández & Procacci's SCUB [2], the new SCUBs also improve the Lovász Local Lemma.…”
Section: Introductionmentioning
confidence: 82%
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“…There is a deep connection between the Lovász Local Lemma [5] and the partition function of the abstract polymer system at negative real fugacity discovered by Scott & Sokal [13]. As already done for Fernández & Procacci's SCUB [2], the new SCUBs also improve the Lovász Local Lemma.…”
Section: Introductionmentioning
confidence: 82%
“…The proof is inductiveà la Dobrushin [4] and an extension of the homogeneous version stated in [13,Corollary 5.7]. The induction over the cardinality of finite subsets Λ of P and an ordering of the polymers ensures a bound of ϕ γ Λ without ever considering I (γ) \ Λ = ∅.…”
Section: Discrepancy Between the Known Scubsmentioning
confidence: 99%
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“…By slight abuse of terminology, if Ψ ⊂ C and Ω = Ψ n then Ω-stable polynomials will also be referred to as Ψ-stable. In physics [54,56] it is useful to distinguish between hard-core pair interactions (subject to constraints, e.g. the maximum degree of a graph) and soft-core pair interactions (essentially constraint-free).…”
Section: Introductionmentioning
confidence: 99%