Hierarchical exchangeability of pure states in mean field spin glass models

TL;DR

This paper proves hierarchical exchangeability of pure states in mean field spin glass models, supporting the M"ezard-Parisi ansatz and advancing understanding of the structure of Gibbs measures in these complex systems.

Contribution

It establishes hierarchical exchangeability of pure states in mean field spin glasses, connecting the Ghirlanda-Guerra identities with recent representation results.

Findings

Hierarchical exchangeability of pure states proven

Supports the M"ezard-Parisi ansatz structure

Identifies the key missing property for full ansatz validation

Abstract

The main result in this paper is motivated by the M\'ezard-Parisi ansatz which predicts a very special structure for the distribution of spins in diluted mean field spin glass models, such as the random K-sat model. Using the fact that one can safely assume the validity of the Ghirlanda-Guerra identities in these models, we prove hierarchical exchangeability of pure states for the asymptotic Gibbs measures, which allows us to apply a representation result for hierarchically exchangeable arrays recently proved in arXiv:1301.1259. Comparing this representation with the predictions of the M\'ezard-Parisi ansatz, one can see that the key property still missing is that the multi-overlaps between pure states depend only on their overlaps.