{
  "id": "1612.08809",
  "version": "v1",
  "published": "2016-12-28T06:34:20.000Z",
  "updated": "2016-12-28T06:34:20.000Z",
  "title": "Mean-field bound on the 1-arm exponent for Ising ferromagnets in high dimensions",
  "authors": [
    "Satoshi Handa",
    "Markus Heydenreich",
    "Akira Sakai"
  ],
  "comment": "14 pages, 1 figure",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "The 1-arm exponent $\\rho$ for the ferromagnetic Ising model on $\\mathbb{Z}^d$ is the critical exponent that describes how fast the critical 1-spin expectation at the center of the ball of radius $r$ surrounded by plus spins decays in powers of $r$. Suppose that the spin-spin coupling $J$ is translation-invariant, $\\mathbb{Z}^d$-symmetric and finite-range. Using the random-current representation and assuming the anomalous dimension $\\eta=0$, we show that the optimal mean-field bound $\\rho\\le1$ holds for all dimensions $d>4$. This significantly improves a bound previously obtained by a hyperscaling inequality.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-12-28T06:34:20.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "high dimensions",
      "ising ferromagnets",
      "optimal mean-field bound",
      "plus spins decays",
      "random-current representation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}