期刊
JOURNAL OF STATISTICAL PHYSICS
卷 118, 期 5-6, 页码 1151-1261出版社
SPRINGER
DOI: 10.1007/s10955-004-2055-4
关键词
graph; lattice gas; hard-core interaction; independent-set polynomial; polymer expansion; cluster expansion; Mayer expansion; Lovasz local lemma; probabilistic method
We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz 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 Lovasz 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 Lovasz 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.
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