Some remarks on p-spin interaction models in a random field

We present some calculations for the thermodynamic behavior of mean-field ferromagnetic p-spin interaction models in the presence of quenched random fields. For both Ising and spherical spin variables, we use the law of large numbers, without recourse to the replica trick, to obtain a free-energy functional in terms of the order parameter and an extra non-ordering density. In the spherical limit, we show that the transition is continuous for p=2, but turns into first order for p⩾3, regardless of the probability distribution of the random fields. In the Ising case, for p=2, we recover previously known results. The free-energy functional obtained in this treatment can be used as a starting point for a dynamical study of these models.