Modified log-Sobolev inequalities, concentration bounds and uniqueness of Gibbs measures

TL;DR

This paper establishes a link between modified log-Sobolev inequalities and the uniqueness of Gibbs measures, showing that certain concentration bounds imply uniqueness and that non-uniqueness regimes violate these inequalities.

Contribution

It demonstrates that modified log-Sobolev inequalities imply uniqueness of translation-invariant Gibbs measures and characterizes regimes where these inequalities cannot hold.

Findings

Uniqueness of Gibbs measures follows from concentration bounds.

Modified log-Sobolev inequalities are not satisfied in non-uniqueness regimes.

Exponential convergence of birth-and-death dynamics is linked to these inequalities.

Abstract

We prove that there is only one translation-invariant Gibbsian point process w.r.t. to a chosen interaction if any of them satisfies a certain bound related to concentration-of-measure. This concentration-of-measure bound is e.g. fulfilled if a corresponding modified logarithmic Sobolev inequality holds. In particular, for natural examples with non-uniqueness regimes, a modified logarithmic Sobolev inequality cannot be satisfied. Therefore, in these situations, the free-energy dissipation in related continuous-time birth-and-death dynamics in $\mathbb{R}^d$ is not exponentially fast.