Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

J. R. Chazottes; J. Moles; F. Redig; E. Ugalde

doi:10.1007/s10955-020-02658-1

Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

J. R. Chazottes
, J. Moles
, F. Redig
, E. Ugalde

CNRS
Universidad Autónoma de San Luis Potosi
Delft Institute of Applied Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd where d≥ 1 and S is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.

Original language	English
Pages (from-to)	2131-2149
Number of pages	19
Journal	Journal of Statistical Physics
Volume	181
Issue number	6
DOIs	https://doi.org/10.1007/s10955-020-02658-1
Publication status	Published - 1 Dec 2020

Keywords

Blowing-up property
Concentration inequalities
Equilibrium states
Hamming distance
Large deviations
Relative entropy

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10.1007/s10955-020-02658-1

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abstract = "We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd where d≥ 1 and S is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.",

keywords = "Blowing-up property, Concentration inequalities, Equilibrium states, Hamming distance, Large deviations, Relative entropy",

author = "Chazottes, \{J. R.\} and J. Moles and F. Redig and E. Ugalde",

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AU - Redig, F.

AU - Ugalde, E.

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N2 - We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd where d≥ 1 and S is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.

AB - We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space SZd where d≥ 1 and S is a finite set. We prove that if an equilibrium state for a shift-invariant uniformly summable potential satisfies a Gaussian concentration bound, then it is unique. Equivalently, if there exist several equilibrium states for a potential, none of them can satisfy such a bound.

KW - Blowing-up property

KW - Concentration inequalities

KW - Equilibrium states

KW - Hamming distance

KW - Large deviations

KW - Relative entropy

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DO - 10.1007/s10955-020-02658-1

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JO - Journal of Statistical Physics

JF - Journal of Statistical Physics

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Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

Abstract

Keywords

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