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

Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-neighbor Interactions on Trees