The repulsive lattice gas, the independent-set polynomial, and the Lov\'asz local lemma

TL;DR

This paper explores the deep connection between the repulsive lattice gas model in statistical mechanics and the Lovász local lemma in combinatorics, providing new insights and generalizations for dependency graphs and probabilistic bounds.

Contribution

It establishes an equivalence between the Lovász local lemma and the nonvanishing of the independent-set polynomial, extending the lemma to soft dependencies and analyzing properties of the lattice gas partition function.

Findings

The Lovász local lemma holds iff the independent-set polynomial is nonvanishing in a specific polydisc.

The proof of the Lovász local lemma corresponds to a simple inductive argument for polynomial nonvanishing.

General properties of the partition function of a repulsive lattice gas are derived, including the alternating-sign property.

Abstract

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 and explicitly by Dobrushin. 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.