Symmetric Langevin spin glass dynamics. (English) Zbl 0954.60031

Summary: We study the asymptotic behavior of symmetric spin glass dynamics in the Sherrington-Kirkpatrick model as proposed by Sompolinsky-Zippelius. We prove that the averaged law of the empirical measure on the path space of these dynamics satisfies a large deviation upper bound in the high temperature regime. We study the rate function which governs this large deviation upper bound and prove that it achieves its minimum value at a unique probability measure \(Q\) which is not Markovian. We deduce an averaged and a quenched law of large numbers. We then study the evolution of the Gibbs measure of a spin glass under Sompolinsky-Zippelius dynamics. We also prove a large deviation upper bound for the law of the empirical measure and describe the asymptotic behavior of a spin on path space under this dynamic in the high temperature regime.


60F10 Large deviations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60K35 Interacting random processes; statistical mechanics type models; percolation theory
82C44 Dynamics of disordered systems (random Ising systems, etc.) in time-dependent statistical mechanics
82C31 Stochastic methods (Fokker-Planck, Langevin, etc.) applied to problems in time-dependent statistical mechanics
82C22 Interacting particle systems in time-dependent statistical mechanics
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