On Dependency Graphs and the Lattice Gas

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 [28] and explicitly by Dobrushin [12, 13]. We also present a generalization of the Lovász Local Lemma that allows for ‘soft’ dependencies. The paper aims to provide an accessible discussion of these results, which are drawn from a longer paper [26] that has appeared elsewhere.