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

Abstract

We prove universality of the Ghirlanda-Guerra identities and spin distributions in the mixed $p$-spin models. The assumption for the universality of the identities requires exactly that the coupling constants have zero means and finite variances, and the result applies to the Sherrington-Kirkpatrick model. As an application, we obtain weakly convergent universality of spin distributions in the generic $p$-spin models under the condition of two matching moments. In particular, certain identities for 3-overlaps and 4-overlaps under the Gaussian disorder follow. Under the stronger mode of total variation convergence, we find that universality of spin distributions in the mixed $p$-spin models holds if mild dilution of connectivity by the Viana-Bray diluted spin glass Hamiltonians is present and the first three moments of coupling constants in the mixed $p$-spin Hamiltonians match. These universality results are in stark contrast to the characterization of spin distributions in the undiluted mixed $p$-spin models, which is known up to now that four matching moments are required in general.

26 pages.To appear in Annales de l'Institut Henri Poincaré, Probabilités et Statistiques

References in corpus (2)

• Spin glass models from the point of view of spin distributions
• A simple invariance theorem