Witness trees in the Moser-Tardos algorithmic Lov\'asz Local Lemma and Penrose trees in the hard core lattice gas

TL;DR

This paper reveals a deep connection between the Moser-Tardos algorithmic Lovász Local Lemma and Penrose trees in the hard core lattice gas, showing that algorithm success aligns with cluster expansion convergence.

Contribution

It establishes an equivalence between witness trees in the Moser-Tardos algorithm and Penrose trees in statistical mechanics, linking algorithmic success to physical model convergence.

Findings

Witness trees correspond to Penrose trees in cluster expansion.

Algorithm success is guaranteed when the cluster expansion converges.

Provides a unified view connecting combinatorics and statistical mechanics.

Abstract

We point out a close connection between the Moser-Tardos algorithmic version of the Lov\'asz 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.