Product type potential on the $XY$ model: selection of maximizing probability and a large deviation principle

Given an interval $[a,b]$ the associated $XY$ model is the space $\mathrm{\Omega }=[a,b{]}^{\mathbb{N}}$ with an a priori probability $\nu$ on the state space $[a,b]$. We will present here the case of the product type potential on the $XY$ model and in this setting we can show the explicit expression of the equilibrium probability. We will also consider questions about Ergodic Optimization, maximizing probabilities, subactions and we will show selection of a maximizing probability, when temperature goes to zero. Finally we show a large deviation principle when temperature goes to zero and we present an explicit expression for the deviation function.