Probability Theory and Related Fields

, Volume 102, Issue 4, pp 455-509

First online:

Large deviations for Langevin spin glass dynamics

  • G. B. ArousAffiliated withURA 762, CNRS, DMI, Ecole Normale Superieure
  • , A. GuionnetAffiliated withURA 743, CNRS, Université de Paris Sud

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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.

Mathematics Subject Classification

60F10 60H10 60K35 82C44