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 $d\ge 1$ 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.