## The Annals of Probability

### Symmetric Langevin spin glass dynamics

#### Abstract

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.

#### Article information

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

Dates
First available in Project Euclid: 18 June 2002

http://projecteuclid.org/euclid.aop/1024404517

Digital Object Identifier
doi:10.1214/aop/1024404517

Mathematical Reviews number (MathSciNet)
MR1457623

Zentralblatt MATH identifier
0954.60031

#### Citation

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

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