ON THE GENERAL ONE-DIMENSIONAL XY MODEL: POSITIVE AND ZERO TEMPERATURE, SELECTION AND NON-SELECTION

Abstract

We consider (M, d) a connected and compact manifold and we denote by [Formula: see text] the Bernoulli space Mℤ. The analogous problem on the half-line ℕ is also considered. Let [Formula: see text] be an observable. Given a temperature T, we analyze the main properties of the Gibbs state [Formula: see text]. In order to do our analysis, we consider the Ruelle operator associated to [Formula: see text], and we get in this procedure the main eigenfunction [Formula: see text]. Later, we analyze selection problems when the temperature goes to zero: (a) existence, or not, of the limit [Formula: see text], a question about selection of subactions, and, (b) existence, or not, of the limit [Formula: see text], a question about selection of measures. The existence of subactions and other properties of Ergodic Optimization are also considered. The case where the potential depends just on the coordinates (x0, x1) is carefully analyzed. We show, in this case, and under suitable hypotheses, a Large Deviation Principle, when T → 0, graph properties, etc. Finally, we will present in detail a result due to van Enter and Ruszel, where the authors show, for a particular example of potential A, that the selection of measure [Formula: see text] in this case, does not happen.