Solving the spherical p -spin model with the cavity method: equivalence with the replica results

The spherical p-spin is a fundamental model for glassy physics, thanks to its analytical solution achievable via the replica method. Unfortunately, the replica method has some drawbacks: it is very hard to apply to diluted models and the assumptions beyond it are not immediately clear. Both drawbacks can be overcome by the use of the cavity method; however, this needs to be applied with care to spherical models. Here, we show how to write the cavity equations for spherical p-spin models, both in the replica symmetric (RS) ansatz (corresponding to belief propagation) and in the one-step replica-symmetry-breaking (1RSB) ansatz (corresponding to survey propagation). The cavity equations can be solved by a Gaussian RS and multivariate Gaussian 1RSB ansatz for the distribution of the cavity fields. We compute the free energy in both ansatzes and check that the results are identical to the replica computation, predicting a phase transition to a 1RSB phase at low temperatures. The advantages of solving the model with the cavity method are many. The physical meaning of the ansatz for the cavity marginals is very clear. The cavity method works directly with the distribution of local quantities, which allows us to generalize the method to diluted graphs. What we are presenting here is the first step towards the solution of the diluted version of the spherical p-spin model, which is a fundamental model in the theory of random lasers and interesting per se as an easier-to-simulate version of the classical fully connected p-spin model.