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2014 The approach of Otto-Reznikoff revisited
Georg Menz
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Electron. J. Probab. 19: 1-27 (2014). DOI: 10.1214/EJP.v19-3418

Abstract

In this article we consider a lattice system of unbounded continuous spins. Otto and 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 which a more detailed analysis based on two new ingredients is needed. The two new ingredients are a covariance estimate and a uniform moment estimate. We additionally provide a comparison principle for covariances showing that the correlations of a conditioned Gibbs measure are controlled by the correlations of the original Gibbs measure with ferromagnetic interaction. This comparison principle simplifies the verification of the hypotheses of the main result. As an application of the main result we show how sufficient algebraic decay of correlations yields the uniqueness of the infinite-volume Gibbs measure, generalizing a result of Yoshida from finite-range to infinite-range interaction.

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Georg Menz. "The approach of Otto-Reznikoff revisited." Electron. J. Probab. 19 1 - 27, 2014. https://doi.org/10.1214/EJP.v19-3418

Information

Accepted: 6 November 2014; Published: 2014
First available in Project Euclid: 4 June 2016

zbMATH: 1307.60145
MathSciNet: MR3275859
Digital Object Identifier: 10.1214/EJP.v19-3418

Subjects:
Primary: 60K35
Secondary: 82B20 , 82C26

Keywords: continuous spin , decay of correlations , lattice systems , Logarithmic Sobolev inequality

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