The geometry of the Gibbs measure of pure spherical spin glasses

Abstract

We analyze the statics for pure $p$-spin spherical spin glass models with $p\geq3$, at low enough temperature. With $F_{N,\beta}$ denoting the free energy, we compute the second order (logarithmic) term of $NF_{N,\beta}$ and prove that, for an appropriate centering $c_{N,\beta}$, $NF_{N,\beta}-c_{N,\beta}$ is a tight sequence. We establish the absence of temperature chaos and analyze the transition rate to disorder chaos of the Gibbs measure and ground state. Those results follow from the following geometric picture we prove for the Gibbs measure, of interest by itself: asymptotically, the measure splits into infinitesimal spherical `bands' centered at deep minima, playing the role of so-called `pure states'. For the pure models, the latter makes precise the so-called picture of `many valleys separated by high mountains' and significant parts of the TAP analysis from the physics literature

On the energy landscape of spherical spin glasses

“…Of course, inside each connected component of L(η) there exists at least one local maxima of H N . As we will see in the next section, this fact combined with Theorem 6 below provides a different proof and extends the results of Subag [15] about the orthogonality of critical points in the pure p-spin model (See Remark 2). Another advantage of our approach is that it also allows to establish Theorems 1 and 2 in the setting of the mixed even p-spin models with Ising-spin configuration space following an identical argument.…”

“…While it is in general very difficult to compare the values of two extrema Gaussian fields, it turns out that the current case is achievable and the set A depends closely on the Parisi measure ρ P . The above strategy is different from the approaches used in the recent studies of the landscape of spherical p-spin models, especially those connected to the complexity of such functions [3,2,14,15,16].…”

The TAP–Plefka Variational Principle for the Spherical SK Model

“…The proof of Theorem 1 splits into a proof of a lower bound and a proof of an upper bound for the partition function Z N (β, h N ). Both are based on recentering the Hamiltonian around magnetizations m of potential pure states (a similar recentering has been used by TAP [27], Bolthausen [6] and Subag [22]). In general, recentering around a given m gives rise to an effective external field for the recentered Hamiltonian.…”

Approximate Ultrametricity for Random Measures and Applications to Spin Glasses

“…: After this paper was first submitted, progress was made by Subag [32] in the setting of spherical models. In particular, in [32], building on the work of [4,5,31,33], Subag shows approximate ultrametricity for pure p-spin models ( .t / Ď 2 t p ) at large values ofˇin a quantitative sense and in the process obtains several sharp results regarding fluctuations of free energies and temperature chaos. (In these models, consists of exactly two atoms.)…”

“…Before turning to these results, we note here that after the submission of this paper, progress was made by Subag [32] regarding approximate ultrametricity for spherical p-spin models. For the definition of these models and relationship between the results of [32] and those of this paper, see Section 2.2. We now turn to the results.…”