  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.
