Construction of pure states in mean field models for spin glasses

Abstract

If a mean field model for spin glasses is generic in the sense that it satisfies the extended Ghirlanda-Guerra identities, and if the law of the overlaps has a point mass at the largest point q * of its support, we prove that one can decompose the configuration space into a sequence of sets (A k ) such that, generically, the overlap of two configurations is equal to q * if and only if they belong to the same set A k . For the study of the overlaps each set A k can be replaced by a single point. Combining this with a recent result of Panchenko (A connection between Ghirlanda-Guerra identities and ultrametricity. Ann Probab (2008, to appear)) this proves that if the overlaps take only finitely many values, ultrametricity occurs. We give an elementary, self-contained proof of this result based on simple inequalities and an averaging argument.

Thouless–Anderson–Palmer equations for generic $p$-spin glasses

“…We recall the precise definition of these sets in Appendix A.1. A similar decomposition was obtained by Talagrand in [14].…”

Introduction to the SK model

“…Motivated by this development, it was shown in Panchenko [69] under the same technical assumption that the Ghirlanda-Guerra identities also imply ultrametricity (an elementary proof was later found in Panchenko [71]). Another approach was given by Talagrand in [94]. However, according to physicists, at low temperature the overlap it not expected to take finitely many values in the thermodynamic limit, so all these result were not directly applicable to the SK model and could not be used to prove the Parisi formula.…”

Universality of Ghirlanda–Guerra identities and spin distributions in mixed $p$-spin models

“…It is used to obtain certain integration by parts involving the internal energy densities N −1 H N,ξ , which is central to the derivation of the identities. As immediate consequences of the universality of the identities, we obtain, in the particular case of generic p-spin models (see (3.1) for the definition), universality of the Gibbs measures of pure states [15] in the sense of Talagrand's construction [26,Theorem 2.4] and universality of the Parisi ultrametricity by Panchenko's theorem [17,Theorem 1]. It is proven in [1], among other things, universality of the Parisi ultrametricity in the mixed p-spin models.…”

“…The proof of (3.16) with m = 3 is similar. Now with the disorder η N chosen at the beginning of this proof, we work with the interpolating coupling constants 26) and write the corresponding Gibbs expectations as · N,t . By the choice of η N i , the arguments in Step 1 and Step 2 (see (3.18), (3.21) and (3.23) in particular) can be modified slightly to show that, for all F : Σ n N → [−1, 1], k ∈ {1, · · · , N } 2 and T ∈ [0, 1], (3.27) where E N,t (F ) converges to zero as N → ∞ uniform in t ∈ (0, 1].…”