Magnetization Densities as Replica Parameters: The Dilute Ferromagnet

In this paper we compute exactly the ground state energy and entropy of the dilute ferromagnetic Ising model. The two thermodynamic quantities are also computed when a magnetic field with random locations is present. The result is reached in the replica approach frame by a class of replica order parameters introduced by Monasson. The strategy is first illustrated considering the SK model, for which we will show the complete equivalence with the standard replica approach. Then, we apply to the diluted ferromagnetic Ising model with a random located magnetic field, which is mapped into a Potts model. This formalism can be, in principle, applied to all random systems, and we believe that it could be of help in many other contexts.