Gaussian concentration and uniqueness of equilibrium states in lattice systems

We consider equilibrium states (that is, shift-invariant Gibbs measures) on the configuration space $S^{\mathbb{Z}^d}$ where $d\geq 1$ 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.
Comment: 24 pages. Accepted for publication in J. Stat. Phys. (2020). Some typos have been corrected. Proposition 2.2 has been strengthened: if a Gaussian concentration bound holds then the measure is mixing, not only ergodic