Exact Free Energy of Dilute p-spin Model.

In this paper we evaluate the free energy of the dilute p-spin model at finite temperature. Franz et al gave the first step to rigourize the replica solution for the dilute p-spin model by proving that the solution given by the Parisi replica scheme is a lower bound for p even. In this paper we also show that the replica symmetric solution is a lower bound to the free energy of the dilute p-spin model for the case of p odd in some region of the temperature and $p$-spin parameters.