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Probability Theory and Related Fields

, Volume 156, Issue 1–2, pp 375–413 | Cite as

Random overlap structures: properties and applications to spin glasses

  • Louis-Pierre Arguin
  • Sourav Chatterjee
Article

Abstract

Random overlap structures (ROSt’s) are random elements on the space of probability measures on the unit ball of a Hilbert space, where two measures are identified if they differ by an isometry. In spin glasses, they arise as natural limits of Gibbs measures under the appropriate algebra of functions. We prove that the so called ‘cavity mapping’ on the space of ROSt’s is continuous, leading to a proof of the stochastic stability conjecture for the limiting Gibbs measures of a large class of spin glass models. Similar arguments yield the proofs of a number of other properties of ROSt’s that may be useful in future attempts at proving the ultrametricity conjecture. Lastly, assuming that the ultrametricity conjecture holds, the setup yields a constructive proof of the Parisi formula for the free energy of the Sherrington–Kirkpatrick model by making rigorous a heuristic of Aizenman, Sims and Starr.

Keywords

Probability measures on Hilbert spaces Spin glasses Stochastic stability Ultrametricity Parisi formula 

Mathematics Subject Classification

60G15 60G57 82B44 

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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  1. 1.Département de Mathématiques et StatistiqueUniversité de MontréalMontréalCanada
  2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA

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