Electronic Journal of Probability

The approach of Otto-Reznikoff revisited

Georg Menz

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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.

Article information

Source
Electron. J. Probab., Volume 19 (2014), paper no. 107, 27 pp.

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

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465065749

Digital Object Identifier
doi:10.1214/EJP.v19-3418

Mathematical Reviews number (MathSciNet)
MR3275859

Zentralblatt MATH identifier
1307.60145

Subjects
Primary: 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43]
Secondary: 82B20: Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs 82C26: Dynamic and nonequilibrium phase transitions (general)

Keywords
lattice systems continuous spin logarithmic Sobolev inequality decay of correlations

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Menz, Georg. The approach of Otto-Reznikoff revisited. Electron. J. Probab. 19 (2014), paper no. 107, 27 pp. doi:10.1214/EJP.v19-3418. https://projecteuclid.org/euclid.ejp/1465065749


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