  Given an interval $[a,b]$ the associated $X\,Y$ model is the space
$\Omega=[a,b]^\mathbb{N}$ 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 $X\,Y$
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.
