Metastability and maximal-entropy joinings of Gibbs measures on finitely-generated groups

We prove a metastability result for finitary microstates which are good models for a Gibbs measure for a nearest-neighbor interaction on a finitely-generated group. This is used to show that any maximal-entropy joining of two such Gibbs states is a relative product over the tail $\sigma$-algebra, except in degenerate cases. We also use results on extremal cuts of random graphs to further investigate optimal self-joinings of the Ising model on a free group.