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.
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L.-P. Arguin held a postdoctoral position at Courant Institute during this work. He was supported by the NSF grant DMS-0604869 and partially by the Hausdorff Center for Mathematics, Bonn. S. Chatterjee’s research was partially supported by NSF grants DMS-0707054 and DMS-1005312, and a Sloan Research Fellowship.
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Arguin, LP., Chatterjee, S. Random overlap structures: properties and applications to spin glasses. Probab. Theory Relat. Fields 156, 375–413 (2013). https://doi.org/10.1007/s00440-012-0431-6
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DOI: https://doi.org/10.1007/s00440-012-0431-6