Gibbs measures and Glauber dynamics on trees

The Ising model has two a priori different notions of equilibrium state: Gibbs states (which satisfy the DLR equations) and Glauber-invariant states (which are invariant under a natural dynamics). On the integer lattice these notions have long been known to be equivalent for shift-invariant states. Moreover, Holley showed in 1971 that any shift-invariant state converges weakly to the set of Gibbs states when evolved under Glauber dynamics. I will show how these results can be extended to infinite regular trees by defining a new notion of free energy density in the framework of sofic entropy theory.

Link