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Large deviations for Langevin spin glass dynamics

Probability Theory and Related Fields volume 102, pages 455–509 (1995)Cite this article

Summary

We study the asymptotic behaviour of asymmetrical spin glass dynamics in a Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the annealed law of the empirical measure on path space of these dynamics satisfy a large deviation principle in the high temperature regime. We study the rate function of this large deviation principle and prove that it achieves its minimum value at a unique probability measureQ which is not markovian. We deduce that the quenched law of the empirical measure converges to δ Q . Extending then the preceeding results to replicated dynamics, we investigate the quenched behavior of a single spin. We get quenched convergence toQ in the case of a symmetric initial law and even potential for the free spin.

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Authors and Affiliations

  1. URA 762, CNRS, DMI, Ecole Normale Superieure, F-75230, Paris, France

    G. B. Arous

  2. URA 743, CNRS, Université de Paris Sud, Bat. 425, F-91405, Orsay, France

    A. Guionnet

Authors
  1. G. B. Arous
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  2. A. Guionnet
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Arous, G.B., Guionnet, A. Large deviations for Langevin spin glass dynamics. Probab. Th. Rel. Fields 102, 455–509 (1995). https://doi.org/10.1007/BF01198846

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  • DOI: https://doi.org/10.1007/BF01198846

Mathematics Subject Classification

  • 60F10
  • 60H10
  • 60K35
  • 82C44