Selection of measure and a Large Deviation Principle for the general XY model

Abstract

We consider $(M,d)$ a connected and compact manifold and we denote by $X$ the Bernoulli space $M^{\mathbb{N}}$. The shift acting on $X$ is denoted by $\sigma$. We analyze 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 ${\mu }_c:=h_c{\nu }_c$, where $h_c$ is the eigenfunction, and, ${\nu }_c$ is the eigenmeasure of the Ruelle operator associated to $cf$. We are going to prove that any measure selected by ${\mu }_c$, as $c\to +\infty$, is a maximizing measure for $f$. We also show, when the maximizing probability measure is unique, that it is true a Large Deviation Principle, with the deviation function $R_{+}^{\infty }={\sum }_{j=0}^{\infty }{R}_{+}({\sigma }^f)$, where $R_{+}:=\beta (f)+V\circ \sigma -V-f$, and, $V$ is any calibrated subaction.

References in corpus (2)

• On the zero-temperature limit of Gibbs states
• Selection of measures for a potential with two maxima at the zero temperature limit

Cited by in corpus (1)

• The Analyticity of a Generalized Ruelle's Operator