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Abstract 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 typically fails to be equilibrium as long as it is not theonlyGibbs state. For every temperature between the uniqueness and reconstruction thresholds a typical sofic approximation gives this state finite but non-maximal pressure, and for every lower temperature the pressure is non-maximal overeverysofic approximation. We also show that, for more general interactions on sofic groups, the local on average limit of Gibbs states over a sofic approximation Σ, if it exists, is a mixture of Σ-equilibrium states. We use this to show that the plus- and minus-boundary-condition Ising states are Σ-equilibrium if Σ 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. 

We extend results of Richard Holley beyond the integer lattice to a large class of countable groups which includes free groups and all amenable groups: for nearest-neighbor interactions on the Cayley graphs of such groups, we show that a shift-invariant measure is Gibbs if and only if it is Glauber-invariant. Moreover, any shift-invariant measure converges weakly to the set of Gibbs measures when evolved under the corresponding Glauber dynamics. These results are proven using a notion of free energy density relative to a sofic approximation by homomorphisms, which avoids the boundary problems which appear when applying a standard free energy method in a nonamenable setting. We also show that any measure which minimizes this free energy density is Gibbs. 

Abstract The f -invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen, who first used it to show that two finite-entropy Bernoulli shifts over a finitely generated free group can be isomorphic only if their base measures have the same Shannon entropy. Bowen also showed that the f -invariant is a variant of sofic entropy; in particular, it is the exponential growth rate of the expected number of good models over a uniform random homomorphism. In this paper we present an analogous formula for the relative f -invariant and use it to prove a formula for the exponential growth rate of the expected number of good models over a random sofic approximation which is a type of stochastic block model.