Spin Systems on Bethe Lattices

  • Amin Coja-OghlanEmail author
  • Will Perkins


In an extremely influential paper Mézard and Parisi put forward an analytic but non-rigorous approach called the cavity method for studying spin systems on the Bethe lattice, i.e., the random d-regular graph Mézard and Parisi (Eur Phys J B 20:217–233, 2001). Their technique was based on certain hypotheses; most importantly, that the phase space decomposes into a number of Bethe states that are free from long-range correlations and whose marginals are given by a recurrence called Belief Propagation. In this paper we establish this decomposition rigorously for a very general family of spin systems. In addition, we show that the free energy can be computed from this decomposition. We also derive a variational formula for the free energy. The general results have interesting ramifications on several special cases.



The first author thanks Max Hahn-Klimroth for helpful discussions on the cut metric.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Mathematics InstituteGoethe UniversityFrankfurtGermany
  2. 2.Department of Mathematics, Statistics, and Computer ScienceUniversity of Illinois at ChicagoChicagoUSA

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