Hierarchical exchangeability of pure states in mean field spin glass models

The main result in this paper is motivated by the M,zard-Parisi ansatz which predicts a very special structure for the distribution of spins in diluted mean field spin glass models, such as the random -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 M,zard-Parisi ansatz, one can see that the key property still missing is that the multi-overlaps between pure states depend only on their overlaps.