Selection of measure and a large deviation principle for the general one-dimensional XY model

Abstract

We consider (M, d) a connected and compact manifold and we denote by X the Bernoulli space . The shift acting on X is denoted by σ. We analyse the general XY model, as presented in a recent paper by A.T. Baraviera, L.M. Cioletti, A.O. Lopes, J.Mohr and R.R. Souza. Denote the Gibbs measure by μc : = hcνc, where hc is the eigenfunction and νc is the eigenmeasure of the Ruelle operator associated with cf. We will show that any measure selected by μc, as c → +∞, is a maximizing measure for f. We also prove, when the maximizing probability measure is unique, that a certain large deviation principle holds with the deviation function R∞+ = ∑j = 0∞R+(σj), where R+ : = β( f ) + ○σ − − f and V is any calibrated subaction.