{
  "id": "1403.7268",
  "version": "v2",
  "published": "2014-03-28T01:39:59.000Z",
  "updated": "2014-11-25T21:09:50.000Z",
  "title": "Critical two-point function of the 4-dimensional weakly self-avoiding walk",
  "authors": [
    "Roland Bauerschmidt",
    "David C. Brydges",
    "Gordon Slade"
  ],
  "comment": "26 pages, revised version, will appear in Commun. Math. Phys",
  "categories": [
    "math-ph",
    "math.DS",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We prove $|x|^{-2}$ decay of the critical two-point function for the continuous-time weakly self-avoiding walk on $\\mathbb{Z}^d$, in the upper critical dimension $d=4$. This is a statement that the critical exponent $\\eta$ exists and is equal to zero. Results of this nature have been proved previously for dimensions $d \\geq 5$ using the lace expansion, but the lace expansion does not apply when $d=4$. The proof is based on a rigorous renormalisation group analysis of an exact representation of the continuous-time weakly self-avoiding walk as a supersymmetric field theory. Much of the analysis applies more widely and has been carried out in a previous paper, where an asymptotic formula for the susceptibility is obtained. Here, we show how observables can be incorporated into the analysis to obtain a pointwise asymptotic formula for the critical two-point function. This involves perturbative calculations similar to those familiar in the physics literature, but with error terms controlled rigorously.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-03-28T01:39:59.000Z",
      "comment": "22 pages",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-11-25T21:09:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B28",
      "97K99",
      "37A60"
    ],
    "keywords": [
      "critical two-point function",
      "continuous-time weakly self-avoiding walk",
      "lace expansion",
      "supersymmetric field theory",
      "rigorous renormalisation group analysis"
    ],
    "publication": {
      "doi": "10.1007/s00220-015-2353-5",
      "journal": "Communications in Mathematical Physics",
      "year": 2015,
      "month": "Aug",
      "volume": 338,
      "number": 1,
      "pages": 169
    },
    "note": {
      "typesetting": "TeX",
      "pages": 26,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015CMaPh.338..169B"
    }
  }
}