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Large deviations for Langevin spin glass dynamics


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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Arous, G.B., Guionnet, A. Large deviations for Langevin spin glass dynamics. Probab. Th. Rel. Fields 102, 455–509 (1995).

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Mathematics Subject Classification

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