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Statistical Physics, Optimization, Inference, and Message-Passing AlgorithmsLecture Notes of the Les Houches School of Physics: Special Issue, October 2013$
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Florent Krzakala, Federico Ricci-Tersenghi, Lenka Zdeborova, Riccardo Zecchina, Eric W. Tramel, and Leticia F. Cugliandolo

Print publication date: 2015

Print ISBN-13: 9780198743736

Published to Oxford Scholarship Online: March 2016

DOI: 10.1093/acprof:oso/9780198743736.001.0001

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Replica theory and spin glasses

Replica theory and spin glasses

(p.61) 3 Replica theory and spin glasses
Statistical Physics, Optimization, Inference, and Message-Passing Algorithms

Giorgio Parisi

Oxford University Press

This chapter starts with an introduction to the replica method, a standard method in statistical physics, along with a concrete application to the computation of the eigenvalue distribution of random matrices in the Gaussian orthogonal ensemble. The solution of the Sherrington–Kirkpatrick (SK) disordered spin-glass model is then derived, along with the phenomenon of replica symmetry breaking (RSB). The physical meaning of the RSB is explained. The ultrametricity of the space of pure states emerges as a consequence of the hierarchical RSB scheme. Moreover, it is shown how some low-temperature properties of physical observables can be derived by invoking the stochastic stability principle. Some rigorous results on the SK model are described: the existence of the thermodynamic limit, and the proof of the exactness of the hierarchical RSB solution.

Keywords:   replica method, spin-glass, random matrix, stochastic stability, replica symmetry breaking

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