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

Abstract

In a companion paper we developed the generalized TAP approach for general multispecies spherical mixed p-spin models. In this paper, we use it to compute the limit of the free energy at any temperature for all pure multi-species spherical p-spin models, assuming that the limit exists. The pure multi-species do not satisfy the convexity assumption on the mixture which was crucial in the recent proofs of the Parisi formula for the multi-species Sherrington-Kirkpatrick model by Barra et al. (2015) and Panchenko (2015) and for the multi-species spherical mixed p-spin models by Bates and Sohn (2021).

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

“…ξ(q) = β 2 q s for some s ∈ S p , p ≥ 2), which do not satisfy (H3). The TAP approach executed in [69,70] is analogous to [66,67] in the single-species case (with [67] going beyond the aforementioned [71] to cover all temperatures); that methodology bypasses the Parisi framework of the present paper and works on the assumption that E(F N ) converges as N → ∞. At present, this assumption is not known rigorously beyond the cases considered here and in [9,15].…”

“…Regarding the latter, a trio of works by Subag [69,68,70] appeared shortly after this paper was first released, containing respectively (i) a TAP representation for the free energy of general multi-species spherical models; (ii) an analysis of the critical inverse temperature in such models; and (iii) a formula for the limiting free energy (1.16) in pure models (i.e. ξ(q) = β 2 q s for some s ∈ S p , p ≥ 2), which do not satisfy (H3).…”

Optimization of random high-dimensional functions: Structure and algorithms

“…In the spherical case, TAP β (µ m ) is constant on the sphere and only the Hamiltonian is maximized in (43). For the pure spherical models, this can be used to compute the free energy at any temperature [50], also for multi-species models [51,52]. For earlier results on the TAP representation with the classical Onsager correction see [13,15,17,20,22,26,35,47].…”

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).…”

“…Indeed (12) follows easily once we note that (11) are the critical point equations related to the minimisation of ( 7) and we use Lemma 1.…”