Cavity method for supersymmetry-breaking spin glasses

The spontaneous supersymmetry breaking that takes place in certain spin-glass models signals a particularfragility in the structure of metastable states of such systems. This fragility is due to the presence of at least one marginal mode in the Hessian of the free energy, which makes the states highly susceptible under externalperturbations. The cavity method is a technique that recursively describes the property of a system with N+1 spins in terms of those of a system with N spins. To do so, the cavity method assumes a certain degree ofstability when adding a new spin to the system, i.e., it assumes that for a generic choice of the parameters thereis an one-to-one correspondence between the metastable states of the system with N spins and the metastablestates of the system with N+1 spins. In systems where the supersymmetry is broken such a correspondencedoes not exist, and an alternative formulation of the cavity method must be devised.We introduce a generalizedcavity approach that takes care of this problem and we apply it to the computation of the probability distribution of the local magnetizations in the Sherrington-Kirkpatrick model. Our findings agree with the correctsupersymmetry-breaking result.