### Abstract

We consider a version of Glauber dynamics for a p-spin Sherrington- Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γ_{β,p} > 0, such that for all exponential time scales, exp(γ N), with γ < γ_{β,p} , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β^{2} < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.

Original language | English (US) |
---|---|

Pages (from-to) | 663-695 |

Number of pages | 33 |

Journal | Communications in Mathematical Physics |

Volume | 282 |

Issue number | 3 |

DOIs | |

State | Published - Sep 2008 |

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### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

*Communications in Mathematical Physics*,

*282*(3), 663-695. https://doi.org/10.1007/s00220-008-0565-7

**Universality of the REM for dynamics of mean-field spin glasses.** / Ben Arous, Gerard; Bovier, Anton; Černý, Jiří.

Research output: Contribution to journal › Article

*Communications in Mathematical Physics*, vol. 282, no. 3, pp. 663-695. https://doi.org/10.1007/s00220-008-0565-7

}

TY - JOUR

T1 - Universality of the REM for dynamics of mean-field spin glasses

AU - Ben Arous, Gerard

AU - Bovier, Anton

AU - Černý, Jiří

PY - 2008/9

Y1 - 2008/9

N2 - We consider a version of Glauber dynamics for a p-spin Sherrington- Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γβ,p > 0, such that for all exponential time scales, exp(γ N), with γ < γβ,p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β2 < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.

AB - We consider a version of Glauber dynamics for a p-spin Sherrington- Kirkpatrick model of a spin glass that can be seen as a time change of simple random walk on the N-dimensional hypercube. We show that, for all p ≥ 3 and all inverse temperatures β > 0, there exists a constant γβ,p > 0, such that for all exponential time scales, exp(γ N), with γ < γβ,p , the properly rescaled clock process (time-change process) converges to an α-stable subordinator where α = γ/β2 < 1. Moreover, the dynamics exhibits aging at these time scales with a time-time correlation function converging to the arcsine law of this α-stable subordinator. In other words, up to rescaling, on these time scales (that are shorter than the equilibration time of the system) the dynamics of p-spin models ages in the same way as the REM, and by extension Bouchaud's REM-like trap model, confirming the latter as a universal aging mechanism for a wide range of systems. The SK model (the case p = 2) seems to belong to a different universality class.

UR - http://www.scopus.com/inward/record.url?scp=49049115900&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=49049115900&partnerID=8YFLogxK

U2 - 10.1007/s00220-008-0565-7

DO - 10.1007/s00220-008-0565-7

M3 - Article

AN - SCOPUS:49049115900

VL - 282

SP - 663

EP - 695

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 3

ER -