Relative entropy, Gaussian concentration and uniqueness of equilibrium states

TL;DR

This paper demonstrates that Gaussian concentration bounds in lattice-spin systems imply the positivity of relative entropy density, leading to the uniqueness of translation-invariant Gibbs measures, thus extending previous results with a simpler proof.

Contribution

It establishes a new link between Gaussian concentration bounds and the uniqueness of Gibbs measures in lattice-spin systems, providing a more straightforward proof than prior work.

Findings

Gaussian concentration bounds imply positive relative entropy density

Positivity of entropy density ensures uniqueness of Gibbs measures

Extends previous results with a simplified proof

Abstract

For a general class of lattice-spin systems, we prove that an abstract Gaussian concentration bound implies positivity of lower relative entropy density. As a consequence we obtain uniqueness of translation-invariant Gibbs measures from the Gaussian concentration bound in this general setting. This extends earlier results from [Chazottes-Moles-REdig-Ugalde] with a different and very short proof.