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Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists

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  • Ostilli, M.

Abstract

We review critically the concepts and the applications of Cayley Trees and Bethe Lattices in statistical mechanics in a tentative effort to remove widespread misuse of these simple, but yet important–and different–ideal graphs. We illustrate, in particular, two rigorous techniques to deal with Bethe Lattices, based respectively on self-similarity and on the Kolmogorov consistency theorem, linking the latter with the Cavity and Belief Propagation methods, more known to the physics community.

Suggested Citation

  • Ostilli, M., 2012. "Cayley Trees and Bethe Lattices: A concise analysis for mathematicians and physicists," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(12), pages 3417-3423.
    • Handle: RePEc:eee:phsmap:v:391:y:2012:i:12:p:3417-3423
    DOI: 10.1016/j.physa.2012.01.038
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    Citations

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    Cited by:

    1. Correia, A.D. & Leestmaker, L.L. & Stoof, H.T.C. & Broere, J.J., 2022. "Asymmetric games on networks: Towards an Ising-model representation," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 593(C).
    2. 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).
    3. Akın, Hasan & Ulusoy, Suleyman, 2023. "A new approach to studying the thermodynamic properties of the q-state Potts model on a Cayley tree," Chaos, Solitons & Fractals, Elsevier, vol. 174(C).
    4. Mukhamedov, Farrukh & Khakimov, Otabek, 2016. "Phase transition and chaos: P-adic Potts model on a Cayley tree," Chaos, Solitons & Fractals, Elsevier, vol. 87(C), pages 190-196.
    5. 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).
    6. 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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