Free Energy, Gibbs Measures, and Glauber Dynamics for Nearest-Neighbor Interactions

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.