The repulsive lattice gas, the independent-set polynomial, and the lovász local lemma

Alexander D. Scott, Alan D. Sokal

    Research output: Contribution to journalArticle

    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.

    Original languageEnglish (US)
    Pages (from-to)1151-1261
    Number of pages111
    JournalJournal of Statistical Physics
    Volume118
    Issue number5-6
    DOIs
    StatePublished - Mar 2005

    Fingerprint

    Lattice Gas
    Independent Set
    Lemma
    polynomials
    theorems
    Polydisk
    Polynomial
    gases
    Dependency Graph
    Infinite Graphs
    Combinatorics
    statistical mechanics
    Statistical Mechanics
    Partition Function
    partitions
    Refinement
    Radius
    If and only if
    radii
    Sufficient Conditions

    Keywords

    • Cluster expansion
    • Graph
    • Hard-core interaction
    • Independent-set polynomial
    • Lattice gas
    • Lovász local lemma
    • Mayer expansion
    • Polymer expansion
    • Probabilistic method

    ASJC Scopus subject areas

    • Mathematical Physics
    • Physics and Astronomy(all)
    • Statistical and Nonlinear Physics

    Cite this

    The repulsive lattice gas, the independent-set polynomial, and the lovász local lemma. / Scott, Alexander D.; Sokal, Alan D.

    In: Journal of Statistical Physics, Vol. 118, No. 5-6, 03.2005, p. 1151-1261.

    Research output: Contribution to journalArticle

    Scott, Alexander D. ; Sokal, Alan D. / The repulsive lattice gas, the independent-set polynomial, and the lovász local lemma. In: Journal of Statistical Physics. 2005 ; Vol. 118, No. 5-6. pp. 1151-1261.
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