On Marginal Stability in Low Temperature Spherical Spin Glasses

TL;DR

This paper establishes a theoretical link between marginal stability of near-ground states and full replica symmetry breaking in spherical spin glasses at zero temperature, with implications for understanding low-temperature behavior and algorithms.

Contribution

It proves the equivalence of marginal stability and full RSB at zero temperature in spherical spin glasses, and provides new geometric and spectral insights for even and generic models.

Findings

No outlier eigenvalues in the Hessian for even models.

Geometric consequences for low temperature Gibbs measures.

Connection between marginal stability and full RSB established.

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.