Hierarchical exchangeability of pure states in mean field spin glass models

• Published in: Probability theory and related fields Vol. 161; no. 3-4; pp. 619 - 650
• Main Author: Panchenko, Dmitry
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01.04.2015; Springer Nature B.V
• Subjects: Arrays; Asymptotic methods; Asymptotic properties; Dilution; Economics; Exchange; Finance; Insurance; Management; Mathematical analysis; Mathematical and Computational Biology; Mathematical and Computational Physics; Mathematical models; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Probability; Probability theory; Probability Theory and Stochastic Processes; Quantitative Finance; Random variables; Representations; Spin glass; Spin glasses; Statistics for Business; Studies; Texts; Theoretical; Topological manifolds; Spin glasses; 60G09; Diluted models; 82B44; 60K35; Exchangeability
• ISSN: 0178-8051; 1432-2064
• DOI: 10.1007/s00440-014-0555-y

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.