  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.
