Random Overlap Structures: Properties and Applications to Spin Glasses
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 SherringtonKirkpatrick model by making rigorous a heuristic of Aizenman, Sims and Starr.
 Publication:

arXiv eprints
 Pub Date:
 November 2010
 DOI:
 10.48550/arXiv.1011.1823
 arXiv:
 arXiv:1011.1823
 Bibcode:
 2010arXiv1011.1823A
 Keywords:

 Mathematics  Probability;
 Condensed Matter  Disordered Systems and Neural Networks;
 60K35;
 82B44;
 60B20;
 60G57
 EPrint:
 35 pages, final version accepted for publication