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**Replica symmetry breaking and exponential inequalities for the Sherrington-Kirkpatrick model.**
*(English)*
Zbl 1034.82027

Summary: We provide an extremely accurate picture of the Sherrington-Kirkpatrick model in three cases: for high temperature, for large external field and for any temperature greater than or equal to 1 and sufficiently small external field. We describe the system at the level of the central limit theorem, or as physicists would say, at the level of fluctuations around the mean field. We also obtain much more detailed information, in the form of exponential inequalities that express a uniform control over higher order moments. We give a complete, rigorous proof that at the generic point of the predicted low temperature region there is “replica symmetry breaking”, in the sense that the system is unstable with respect to an infinitesimal coupling between two replicas.

### MSC:

82B44 | Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics |

60G15 | Gaussian processes |

60G17 | Sample path properties |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

82D30 | Statistical mechanics of random media, disordered materials (including liquid crystals and spin glasses) |

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\textit{M. Talagrand}, Ann. Probab. 28, No. 3, 1018--1062 (2000; Zbl 1034.82027)

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### References:

[1] | Aizenman, M., Lebowitz, J. and Ruelle, D. (1981). Some rigorous results on the Sherrington-Kirkpatrick model. Comm. Math. Phys. 112 3-20. · Zbl 1108.82312 |

[2] | Comets, F. and Neveu, J. (1995). The Sherrington-Kirkpatrick model of spin-glasses and stochastic calculus: the high temperature case. Comm. Math. Phys. 166 549-564. · Zbl 0811.60098 |

[3] | Mézard, M., Parisi, G. and Virasiro, M. (1987). Spin Glass Theory and Beyond. World Scientific, Singapore. · Zbl 0992.82500 |

[4] | Shcherbina, M. (1997). On the replica-symmetric solution for the Sherrington-Kirkpatrick model. Helv. Phys. Acta. 70 838-853. · Zbl 0899.60096 |

[5] | Talagrand, M. (1998). The Sherrington-Kirkpatrick model: a challenge to mathematicians. Probab. Theory Related Fields 110 109-176. · Zbl 0909.60083 |

[6] | Talagrand, M. (1998). Rigorous results for the Hopfield Model with many patterns. Probab. Theory Related Fields 110 177-286. · Zbl 0897.60041 |

[7] | Talagrand, M. (1998). Huge random structures and mean field models for spin glasses. In Proceedings of the International Congress of Mathematicians, Documenta Math. Extra volume I. · Zbl 0902.60089 |

[8] | Talagrand, M. (1999). Verres de spin et optimisation combinatoire. Séminaire Bourbaki, Astérisque. |

[9] | Talagrand, M. (2000). Probability theory and spin glasses. Unpublished manuscript. · Zbl 1149.60324 |

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