Ground States and Critical Points for Aubry-Mather Theory in Statistical Mechanics

Abstract. We consider statistical mechanics systems defined on a set with some symmetry properties (namely, the set admits an action by a group, which is finitely generated and residually finite). Each of the sites has a real order parameter. We assume that the interaction is ferromagnetic as well as symmetric with respect to the action of the symmetry group as well as invariant under the addition of an integer to the order parameter of all the sites. Given any cocycle of the symmetry group, we prove that there are ground states which satisfy an order condition (known as Birkhoff property) and that are at a bounded distance to the cocycle. The set of ground states we construct for a cocyle, is ordered. Equivalently, this set of ground states forms a (possibly singular) lamination. Furthermore, we show that, given any completely irrational cocycle, either the above lamination consists of a foliation made of a continuous one parameter family of ground states, or, inside any gap of the lamination, there is a well-ordered critical point which is not a ground state. The classical multidimensional Frenkel-Kontorova model with periodic coefficients is included in this framework as a particular case. The symmetry group is just ∫ d and the cocyles are linear functions. In the case that the substrate is one-dimensional and that the interaction is nearest neighbor, our result are the initial results of AubryMather theory. The present framework applies to substrates such as the Bethe lattice and it allows to discuss interactions which are not nearest neighbor.