06-278 Rafael de la Llave, Enrico Valdinoci
Critical points inside the gaps of ground state laminations in statistical mechanics (372K, PostScript) Oct 4, 06
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Abstract. We consider models of interacting particles situated in the points of a discrete set. The state of each particle is determined by a real variable. The particles are interacting with each other and we are interested in ordered ground states. Under the assumption that the set and the interaction are symmetric under the action of a group -- which satisfies some mild assumption --, that the interaction is ferromagnetic, as well as periodic under addition of integers, it was shown in a previous paper that there are many ground states that satisfy an order property called selfconforming or Birkhoff. Under the assumption that the interaction is finite range, we show that either the ground states form a one dimensional family or that there are other Birkhoff critical points which are not ground states. In the case that the set is just a one dimensional lattice and that the interaction is just nearest neighbor, this generalizes Mather's criterion for the existence of invariant circles.

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