Spin glass models from the point of view of spin distributions

TL;DR

This paper introduces a novel functional order parameter for spin glass models using the Aldous-Hoover representation, deriving invariance and self-consistency equations, and explores implications for free energy and overlap distributions.

Contribution

It develops new perturbations and invariance properties for the functional order parameter in spin glass models, linking it to free energy and overlap distributions.

Findings

Derived invariance and self-consistency equations for the order parameter.

Provided explicit formulas for the order parameter under ultrametricity.

Connected the order parameter to Ghirlanda-Guerra identities and Parisi ansatz.

Abstract

In many spin glass models, due to the symmetry among sites, any limiting joint distribution of spins under the annealed Gibbs measure admits the Aldous-Hoover representation encoded by a function $\sigma:[0,1]^4\to\{-1,+1\}$, and one can think of this function as a generic functional order parameter of the model. In a class of diluted models, and in the Sherrington-Kirkpatrick model, we introduce novel perturbations of the Hamiltonian that yield certain invariance and self-consistency equations for this generic functional order parameter and we use these invariance properties to obtain representations for the free energy in terms of $\sigma$. In the setting of the Sherrington-Kirkpatrick model, the self-consistency equations imply that the joint distribution of spins is determined by the joint distributions of the overlaps, and we give an explicit formula for $\sigma$ under the Parisi ultrametricity hypothesis. In addition, we discuss some connections with the Ghirlanda-Guerra identities and stochastic stability and describe the expected Parisi ansatz in the diluted models in terms of $\sigma$.