{
  "id": "1710.07608",
  "version": "v1",
  "published": "2017-10-20T17:04:50.000Z",
  "updated": "2017-10-20T17:04:50.000Z",
  "title": "Translation-Invariant Gibbs States of Ising model: General Setting",
  "authors": [
    "Aran Raoufi"
  ],
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We prove that at any inverse temperature $\\beta$ and on any transitive amenable graph, the automorphism-invariant Gibbs states of the ferromagnetic Ising model are convex combinations of the plus and minus states. This is obtained for a general class of interactions, that is automorphism-invariant and irreducible coupling constants. The proof uses the random current representation of the Ising model. The result is novel when the graph is not $\\mathbb{Z}^d$, or when the graph is $\\mathbb{Z}^d$ but endowed with infinite-range interactions, or even $\\mathbb{Z}^2$ with finite-range interactions. Among the corollaries of this result, we can list continuity of the magnetization at any non-critical temperature, the differentiability of the free energy, and the uniqueness of FK-Ising infinite-volume measures.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-10-20T17:04:50.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "translation-invariant gibbs states",
      "general setting",
      "automorphism-invariant gibbs states",
      "interactions",
      "random current representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}