Abstract: In this talk we revisit a model introduced by Marchetti and Perez in order to understand in a rigorous way a renormalization group approach for the Coulumb gas on the lattice; the model corresponds to a map (depending on a parameter \beta) on the probabilities defined on the integers. Our goal is to provide another approach for the existence of non trivial fixed points of this map when the parameter \beta is smaller than a certain critical value.