The Variational Principle for Gibbs States Fails on Trees

#### R.M. Burton, C.-E. Pfister, J.E. Steif

1995, v.1, №3, 387-406

ABSTRACT

We show how the variational principle for Gibbs States (which says that for the $d$-dimensional cubic lattice, the set of translation invariant Gibbs States is the same as the set of translation invariant measures which maximize entropy minus energy and moreover that this quantity corresponds to the pressure) fails for nearest neighbour finite state statistical mechanical systems on the homogeneous 3-ary tree. Given an interaction there is a unique measure $\mu$ maximizing entropy minus energy, and we give necessary and sufficient conditions so that it is a Gibbs State for that interaction, and that the maximum is equal to the pressure. In the case of a 2-state system, these conditions define a 2-dimensional manifold of the natural 3-dimensional parameter space of interactions, so that generically in the interactions the entropy minus energy for $\mu$ is strictly less than the pressure and $\mu$ is not a Gibbs State (for those parameter values).

Keywords: trees,Gibbs States,variational principle