On Marginal Stability in Low Temperature Spherical Spin Glasses

Abstract: We show marginal stability of near-ground states in spherical spin glasses is equivalent to full replica symmetry breaking at zero temperature near overlap $1$. This connection has long been implicit in the physics literature, which also links marginal stability to the performance of efficient algorithms. For even models, we prove the Hessian has no outlier eigenvalues, and obtain geometric consequences for low temperature Gibbs measures in the case that marginal stability is absent. Our proofs rely on interpolation bounds for vector spin glass models. For generic models, we give another more conceptual argument that full RSB near overlap $1$ implies marginal stability at low temperature.