The approach of Otto-Reznikoff revisited

TL;DR

This paper revisits Otto-Reznikoff's method, weakening assumptions to establish uniform logarithmic Sobolev inequalities from decay of correlations in unbounded spin systems, and applies it to prove uniqueness of Gibbs measures.

Contribution

It improves the Otto-Reznikoff approach by weakening assumptions, introducing new covariance and moment estimates, and providing a comparison principle for covariances.

Findings

Weakening of assumptions needed for LSI from decay of correlations.

Introduction of a new covariance estimate and uniform moment estimate.

Application to prove uniqueness of infinite-volume Gibbs measure.

Abstract

In this article we consider a lattice system of unbounded continuos spins. Otto & Reznikoff used the two-scale approach to show that exponential decay of correlations yields a logarithmic Sobolev inequality (LSI) with uniform constant in the system size. We improve their statement by weakening the assumptions. For the proof a more detailed analysis based on two new ingredients is needed. The two new ingredients are a new basic covariance estimate and a uniform moment estimate. We additionally provide a comparison principle for covariances showing that the correlations for the conditioned Gibbs measures are controlled by the correlations of the original Gibbs measure with ferromagnetic interaction. The latter simplifies the application of the main result. As an application, we show how decay of correlations combined with the uniform LSI yields the uniqueness of the infinite-volume Gibbs measure, generalizing a result of Yoshida form finite-range to infinite-range interaction.