We extend results of R. Holley beyond the integer lattice to a large class of groups which includes free groups. In particular we show that a shift-invariant measure is Gibbs if and only if it is Glauber invariant. Moreover, any shiftinvariant measure converges weakly to the set of Gibbs measures when evolved under Glauber dynamics. These results are proven using a new notion of free energy density relative to a sofic approximation by homomorphisms. Any measure which minimizes free energy density is Gibbs.

The f -invariant is an isomorphism invariant of free-group measure-preserving actions introduced by Lewis Bowen in [Bow10b], where it was used 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. In [Bow10a] Bowen 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 type of stochastic block model.

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