Notes on ferromagnetic diluted P-spin model

In this paper we develop the interpolating cavity field technique for the mean field ferromagnetic p-spin. The model we introduce is a natural extension of the diluted Curie-Weiss model to p>2 spin interactions. Several properties of the free energy are analyzed and, in particular, we show that it recovers the expressions already known for p=2 models and for p>2 fully connected models. Further, as the model lacks criticality, we present extensive numerical simulations to evidence the presence of a first order phase transition and deepen the behavior at the transition line. Overall, a good agreement is obtained among analytical results, numerics and previous works.