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The Bethe lattice spin glass revisited

Author

Listed:
  • M. Mézard

    (Laboratoire de Physique Théorique et Modèles Statistiques, Université Paris Sud, Bâtiment 100, 91405 Orsay Cedex, France)

  • G. Parisi

    (Dipartimento di Fisica, Sezione INFN and Unità INFM, Università di Roma “La Sapienza”, Piazzale Aldo Moro 2, 00185 Rome, Italy)

Abstract

So far the problem of a spin glass on a Bethe lattice has been solved only at the replica symmetric level, which is wrong in the spin glass phase. Because of some technical difficulties, attempts at deriving a replica symmetry breaking solution have been confined to some perturbative regimes, high connectivity lattices or temperature close to the critical temperature. Using the cavity method, we propose a general non perturbative solution of the Bethe lattice spin glass problem at a level of approximation which is equivalent to a one step replica symmetry breaking solution. The results compare well with numerical simulations. The method can be used for many finite connectivity problems appearing in combinatorial optimization.

Suggested Citation

  • M. Mézard & G. Parisi, 2001. "The Bethe lattice spin glass revisited," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 20(2), pages 217-233, March.
    • Handle: RePEc:spr:eurphb:v:20:y:2001:i:2:d:10.1007_pl00011099
    DOI: 10.1007/PL00011099
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    Citations

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    Cited by:

    1. Pierre Paga & Reimer Kuhn, 2014. "Contagion in an interacting economy," Papers 1409.2625, arXiv.org, revised Mar 2015.
    2. Sun, Yi-Fan & Sun, Zheng-Yang, 2019. "Target observation of complex networks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 517(C), pages 233-245.
    3. Barucca, Paolo, 2020. "Spectral density of equitable core–periphery graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 553(C).
    4. Fabrizio Altarelli & Alfredo Braunstein & Luca Dall’Asta & Caterina De Bacco & Silvio Franz, 2015. "The Edge-Disjoint Path Problem on Random Graphs by Message-Passing," PLOS ONE, Public Library of Science, vol. 10(12), pages 1-18, December.
    5. Akın, Hasan, 2024. "Investigation of thermodynamic properties of mixed-spin (2, 1/2) Ising and Blum–Capel models on a Cayley tree," Chaos, Solitons & Fractals, Elsevier, vol. 184(C).
    6. Robson, Dominic T. & Annibale, Alessia & Baas, Andreas C.W., 2022. "Reproducing size distributions of swarms of barchan dunes on Mars and Earth using a mean-field model," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 606(C).
    7. Yilun Shang, 2020. "Multi-Hop Generalized Core Percolation On Complex Networks," Advances in Complex Systems (ACS), World Scientific Publishing Co. Pte. Ltd., vol. 23(01), pages 1-15, March.
    8. Barré, Julien & Gonçalves, Bruno, 2007. "Ensemble inequivalence in random graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 386(1), pages 212-218.
    9. Ostilli, M., 2024. "Exact results for the Ising model on a small-world network," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 641(C).
    10. Caravelli, F., 2023. "On forest expansions for two-body partition functions on tree-like interaction graphs," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 609(C).
    11. Akın, Hasan, 2023. "The classification of disordered phases of mixed spin (2,1/2) Ising model and the chaoticity of the corresponding dynamical system," Chaos, Solitons & Fractals, Elsevier, vol. 167(C).

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