{
  "id": "2105.06103",
  "version": "v2",
  "published": "2021-05-13T06:49:23.000Z",
  "updated": "2021-07-21T06:50:15.000Z",
  "title": "Long-Range Ising Models: Contours, Phase Transitions and Decaying Fields",
  "authors": [
    "Lucas Affonso",
    "Rodrigo Bissacot",
    "Eric O. Endo",
    "Satoshi Handa"
  ],
  "comment": "Preliminary Version, 27 pages",
  "categories": [
    "math-ph",
    "cond-mat.stat-mech",
    "math.MP",
    "math.PR"
  ],
  "abstract": "Inspired by Fr\\\"{o}hlich-Spencer and subsequent authors who introduced the notion of contour for long-range systems, we provide a definition of contour and a direct proof for the phase transition for ferromagnetic long-range Ising models on $\\mathbb{Z}^d$, $d\\geq 2$. The argument, which is based on a multi-scale analysis, works for the sharp region $\\alpha>d$ and improves previous results obtained by Park for $\\alpha>3d+1$, and by Ginibre, Grossmann, and Ruelle for $\\alpha> d+1$, where $\\alpha$ is the power of the coupling constant. The key idea is to avoid a large number of small contours. As an application, we prove the persistence of the phase transition when we add a polynomial decaying magnetic field with power $\\delta>0$ as $h^*|x|^{-\\delta}$, where $h^* >0$. For $d<\\alpha<d+1$, the phase transition occurs when $\\delta>d-\\alpha$, and when $h^*$ is small enough over the critical line $\\delta=d-\\alpha$. For $\\alpha \\geq d+1$, $\\delta>1$ it is enough to prove the phase transition, and for $\\delta=1$ we have to ask $h^*$ small. The natural conjecture is that this region is also sharp for the phase transition problem when we have a decaying field.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2021-07-21T06:50:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B05",
      "82B26",
      "82B20",
      "82Bxx"
    ],
    "keywords": [
      "decaying field",
      "ferromagnetic long-range ising models",
      "polynomial decaying magnetic field",
      "phase transition occurs",
      "phase transition problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}