Structure of 1-RSB asymptotic Gibbs measures in the diluted p-spin models

In this paper we study asymptotic Gibbs measures in the diluted p-spin models in the so called 1-RSB case, when the overlap takes two values $q_*, q^*\in [0,1].$ When the external field is not present and the overlap is not equal to zero, we prove that such asymptotic Gibbs measures are described by the M\'ezard-Parisi ansatz conjectured in [MP]. When the external field is present, we prove that the overlap can not be equal to zero and all 1-RSB asymptotic Gibbs measures are described by the M\'ezard-Parisi ansatz. Finally, we give a characterization of the exceptional case when there is no external field and the smallest overlap value is equal to zero, although it does not go as far as the M\'ezard-Parisi ansatz. Our approach is based on the cavity computations combined with the hierarchical exchangeability of pure states.