{
  "id": "1808.03440",
  "version": "v1",
  "published": "2018-08-10T07:53:47.000Z",
  "updated": "2018-08-10T07:53:47.000Z",
  "title": "Spin systems on Bethe lattices",
  "authors": [
    "Amin Coja-Oghlan",
    "Will Perkins"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.CO",
    "math.MP"
  ],
  "abstract": "In an extremely influential paper Mezard 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 [Eur. Phys. J. B 20 (2001) 217--233]. 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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-08-10T07:53:47.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C80"
    ],
    "keywords": [
      "bethe lattice",
      "free energy",
      "extremely influential paper mezard",
      "phase space decomposes",
      "special cases"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}