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

TL;DR

This paper proves the universality of Ghirlanda-Guerra identities and spin distributions in mixed p-spin models, showing that under certain conditions, these properties are independent of the specific distribution of couplings.

Contribution

It establishes universality results for spin distributions and identities in mixed p-spin models with minimal moment-matching assumptions, including diluted models.

Findings

Universality of Ghirlanda-Guerra identities under zero mean, finite variance couplings.

Weak convergence of spin distributions with two matching moments.

Total variation convergence in diluted models with three matching moments.

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.