Gaussian Concentration and Uniqueness of Equilibrium States in Lattice Systems

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. © 2020, Springer Science Business Media, LLC, part of Springer Nature.
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. © 2020, Springer Science%2bBusiness Media, LLC, part of Springer Nature.

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

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