The Erwin Schrr Odinger International Institute for Mathematical Physics the Variational Principle for Gibbs States Fails on Trees the Variational Principle for Gibbs States Fails on Trees

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 neighbor nite state statistical mechanical systems on the homogeneous 3{ary tree. Given an interaction there is a unique measure maximizing entropy minus energy, and we give necessary and suucient 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 deene a 2{dimensional manifold of the natural 3{dimensional parameter space of interactions, so that generically in the interactions the entropy minus energy for is strictly less than the pressure and is not a Gibbs State (for those parameter values).