### 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.

Original language | English (US) |
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Pages (from-to) | 1367-1422 |

Number of pages | 56 |

Journal | Annals of Probability |

Volume | 25 |

Issue number | 3 |

State | Published - Jul 1997 |

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### Keywords

- Interacting random processes
- Langevin dynamics
- Large deviations
- Statistical mechanics

### ASJC Scopus subject areas

- Mathematics(all)
- Statistics and Probability

### Cite this

*Annals of Probability*,

*25*(3), 1367-1422.

**Symmetric langevin spin glass dynamics.** / Ben Arous, G.; Guionnet, A.

Research output: Contribution to journal › Article

*Annals of Probability*, vol. 25, no. 3, pp. 1367-1422.

}

TY - JOUR

T1 - Symmetric langevin spin glass dynamics

AU - Ben Arous, G.

AU - Guionnet, A.

PY - 1997/7

Y1 - 1997/7

N2 - 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.

AB - 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.

KW - Interacting random processes

KW - Langevin dynamics

KW - Large deviations

KW - Statistical mechanics

UR - http://www.scopus.com/inward/record.url?scp=0040007229&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0040007229&partnerID=8YFLogxK

M3 - Article

VL - 25

SP - 1367

EP - 1422

JO - Annals of Probability

JF - Annals of Probability

SN - 0091-1798

IS - 3

ER -