{
  "id": "1705.08540",
  "version": "v1",
  "published": "2017-05-23T21:22:29.000Z",
  "updated": "2017-05-23T21:22:29.000Z",
  "title": "Critical two-point function for long-range $O(n)$ models below the upper critical dimension",
  "authors": [
    "Martin Lohmann",
    "Gordon Slade",
    "Benjamin C. Wallace"
  ],
  "comment": "30 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider the $n$-component $|\\varphi|^4$ lattice spin model ($n \\ge 1$) and the weakly self-avoiding walk ($n=0$) on $\\mathbb{Z}^d$, in dimensions $d=1,2,3$. We study long-range models based on the fractional Laplacian, with spin-spin interactions or walk step probabilities decaying with distance $r$ as $r^{-(d+\\alpha)}$ with $\\alpha \\in (0,2)$. The upper critical dimension is $d_c=2\\alpha$. For $\\epsilon >0$, and $\\alpha = \\frac 12 (d+\\epsilon)$, the dimension $d=d_c-\\epsilon$ is below the upper critical dimension. For small $\\epsilon$, weak coupling, and all integers $n \\ge 0$, we prove that the two-point function at the critical point decays with distance as $r^{-(d-\\alpha)}$. This \"sticking\" of the critical exponent at its mean-field value was first predicted in the physics literature in 1972. Our proof is based on a rigorous renormalisation group method. The treatment of observables differs from that used in recent work on the nearest-neighbour 4-dimensional case, via our use of a cluster expansion.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-05-23T21:22:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B28",
      "82B27",
      "82B20",
      "60K35"
    ],
    "keywords": [
      "upper critical dimension",
      "critical two-point function",
      "study long-range models",
      "lattice spin model",
      "rigorous renormalisation group method"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 30,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}