  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.
