Witness Trees in the Moser–Tardos Algorithmic Lovász Local Lemma and Penrose Trees in the Hard-Core Lattice Gas

Abstract

We point out a close connection between the Moser–Tardos algorithmic version of the Lovasz local lemma, a central tool in probabilistic combinatorics, and the cluster expansion of the hard-core lattice gas in statistical mechanics. We show that the notion of witness trees given by Moser and Tardos is essentially coincident with that of Penrose trees in the Cluster expansion scheme of the hard-core gas. Such an identification implies that the Moser–Tardos algorithm is successful in a polynomial time if the cluster expansion converges.