Equilibrium and nonequilibrium Gibbs states on sofic groups

Recent work of Barbieri and Meyerovitch has shown that, for very general spin systems indexed by sofic groups, equilibrium (i.e. pressure-maximizing) states are Gibbs. The main goal of this paper is to show that the converse fails in an interesting way: for the Ising model on a free group, the free-boundary state can fail to be equilibrium as long as it is not the only Gibbs state. For every temperature below the uniqueness threshold there exists a sofic approximation which gives this state finite but non-maximal pressure, and below half the uniqueness threshold the pressure is non-maximal over every sofic approximation. We also show that the local limit of Gibbs states over a sofic approximation $\Sigma$, if it exists, is a mixture of $\Sigma$-equilibrium states, and use this to show that the plus- and minus-boundary-condition Ising states are $\Sigma$-equilibrium if $\Sigma$ is any sofic approximation to a free group. Combined with a result of Dembo and Montanari, this implies that these states have the same entropy over every sofic approximation.