The Legendre structure of the TAP complexity for the Ising spin glass

Abstract

We study the complexity of the Thouless-Anderson-Palmer (TAP) free energy for Ising spin glasses with a general mixed p-spin covariance, working with the generalized TAP functional of Chen, Panchenko, and Subag. We formulate three conjectures about the complexity (i.e. number of critical points). First, the annealed complexity is given by the Legendre transform of a variational functional constructed from the Parisi formula subject to a constraint on the overlap mass at zero, thereby establishing a precise link between the enumeration of TAP states and the large-deviation rate function of the partition function. Second, the quenched complexity is governed by the Legendre transform of a closely related functional in which the mass up to -- but not including -- the supremum of the support is constrained. Third, TAP states at any non-equilibrium free-energy level are organized into an ultrametric hierarchy, with ancestor states at other levels appearing only in subexponential number. Using a Kac-Rice computation combined with a supersymmetric ansatz, we establish a lower bound on the annealed complexity that matches the prediction of the first conjecture. We further extend the analysis to a conditional setting in which a hierarchical "skeleton" of ancestors is prescribed, providing additional evidence in support of the second and third conjectures.