Hierarchical exchangeability of pure states in mean field spin glass models

Abstract

The main result in this paper is motivated by the Mézard–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 at positive temperature. 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 Austin and Panchenko in Probab. Theory Relat. Fields 2013. Comparing this representation with the predictions of the Mézard–Parisi ansatz, one can see that the key property still missing is that the multi-overlaps between pure states depend only on their overlaps.