The Annals of Probability

Symmetric Langevin spin glass dynamics

G. Ben Arous and A. Guionnet

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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.

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Ann. Probab., Volume 25, Number 3 (1997), 1367-1422.

First available in Project Euclid: 18 June 2002

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Primary: 60F10: Large deviations 60H10: Stochastic ordinary differential equations [See also 34F05] 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 82C44: Dynamics of disordered systems (random Ising systems, etc.) 82C31: Stochastic methods (Fokker-Planck, Langevin, etc.) [See also 60H10] 82C22: Interacting particle systems [See also 60K35]

Large deviations interacting random processes statistical mechanics Langevin dynamics


Ben Arous, G.; Guionnet, A. Symmetric Langevin spin glass dynamics. Ann. Probab. 25 (1997), no. 3, 1367--1422. doi:10.1214/aop/1024404517.

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