{
  "id": "1403.7424",
  "version": "v1",
  "published": "2014-03-28T15:50:49.000Z",
  "updated": "2014-03-28T15:50:49.000Z",
  "title": "Scaling limits and critical behaviour of the 4-dimensional n-component $|\\varphi|^4$ spin model",
  "authors": [
    "Roland Bauerschmidt",
    "David C. Brydges",
    "Gordon Slade"
  ],
  "comment": "55 pages",
  "categories": [
    "math-ph",
    "math.DS",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider the $n$-component $|\\varphi|^4$ spin model on $\\mathbb{Z}^4$, for all $n \\geq 1$, with small coupling constant. We prove that the susceptibility has a logarithmic correction to mean field scaling, with exponent $\\frac{n+2}{n+8}$ for the logarithm. We also analyse the asymptotic behaviour of the pressure as the critical point is approached, and prove that the specific heat has fractional logarithmic scaling for $n =1,2,3$; double logarithmic scaling for $n=4$; and is bounded when $n>4$. In addition, for the model defined on the $4$-dimensional discrete torus, we prove that the scaling limit as the critical point is approached is a multiple of a Gaussian free field on the continuum torus, whereas, in the subcritical regime, the scaling limit is Gaussian white noise with intensity given by the susceptibility. The proofs are based on a rigorous renormalisation group method in the spirit of Wilson, developed in a companion series of papers to study the 4-dimensional weakly self-avoiding walk, and adapted here to the $|\\varphi|^4$ model.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-28T15:50:49.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B28",
      "97K99",
      "37A60"
    ],
    "keywords": [
      "scaling limit",
      "spin model",
      "critical behaviour",
      "n-component",
      "gaussian white noise"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "doi": "10.1007/s10955-014-1060-5",
      "journal": "Journal of Statistical Physics",
      "year": 2014,
      "month": "Aug",
      "pages": 168
    },
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "inspire": 1287825,
      "adsabs": "2014JSP...tmp..168B"
    }
  }
}