Free energy in multi-species mixed $p$-spin spherical models

Abstract

We prove a Parisi formula for the limiting free energy of multi-species spherical spin glasses with mixed p-spin interactions. The upper bound involves a Guerra-style interpolation and requires a convexity assumption on the model's covariance function. Meanwhile, the lower bound adapts the cavity method of Chen so that it can be combined with the synchronization technique of Panchenko; this part requires no convexity assumption. In order to guarantee that the resulting Parisi formula has a minimizer, we formalize the pairing of synchronization maps with overlap measures so that the constraint set is a compact metric space. This space is not related to the model's spherical structure and can be carried over to other multi-species settings.

TAP approach for multi-species spherical spin glasses II: the free energy of the pure models

“…Combining [36] and [31] yields a variational formula for the ground state energy of the pure bipartite models. As mentioned above, Bates and Sohn proved a Parisi formula for the multi-species spherical mixed p-spin models in [13], assuming the convexity of ξ(x). In [12] they showed it admits a representation analogous to the Crisanti-Sommers formula [25].…”

“…The Gibbs measure at inverse-temperature β > 0 is the random probability measure on S N with density dG N,β dµ (σ) = Z −1 N,β e βHN (σ) . Very recently, Bates and Sohn [13,12] proved a Parisi formula [40,41] for spherical multi-species mixed p-spin models such that the mixture ξ(x) is a convex function. (For the proof of the formula in the single-species case, see [21,38,52,53].)…”

“…One of the main novelties in his paper was the proof that the overlaps R s (σ, σ ′ ) of independent samples from the Gibbs measure are synchronized for different species. The synchronization mechanism of [39] was also used in the spherical setting in [13]. The combination of [8] and [39] establishes the Parisi formula for the multi-species Sherrington-Kirkpatrick model, assuming the convexity of ξ(x).…”

“…This is currently only proven [13] if ξ q (x) is convex on [0, 1] S , but is obviously expected to be true in general. Since for q = 0 the mixture coincides with that of the original model ξ q (x) = ξ(x), under this assumption the limit of the free energy of the pure model exists…”

A remark on the spherical bipartite spin glass

“…Yet these proofs are indirect, as in both cases one obtains a formula for the free energy and then verifies a posteriori (analytically for [2] and numerically [14] for [8]) that it coincides with (3). We just mention that the results in [1] have been recently extended in [10,9] for the complexity and in [5,6] for the free energy (see also [12] for the TAP approach).…”