Probing marginal stability in the spherical $p=2$ model

In this paper we investigate the marginally stable nature of the low-temperature trivial spin glass phase in the spherical $p=2$ spin glass, by perturbing the system with three different kinds of non-linear interactions. In particular, we compare the effect of three additional dense four-body interactions: ferromagnetic couplings, purely disordered couplings and couplings with competing disordered and ferromagnetic interactions. Our study, characterized by the effort to present in a clear and pedagogical way the derivation of all the results, shows that the marginal stability property of the spherical spin glass depends in fact on which kind of perturbation is applied to the system: in general, a certain degree of frustration is needed also in the additional terms in order to induce a transition from a trivial to a non-trivial spin-glass phase. On the contrary, the addition of generic non-frustrated interactions does not destabilize the trivial spin-glass phase.