{
  "id": "1610.08573",
  "version": "v1",
  "published": "2016-10-26T23:30:51.000Z",
  "updated": "2016-10-26T23:30:51.000Z",
  "title": "Four-dimensional weakly self-avoiding walk with contact self-attraction",
  "authors": [
    "Roland Bauerschmidt",
    "Gordon Slade",
    "Benjamin C. Wallace"
  ],
  "comment": "35 pages",
  "categories": [
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "We consider the critical behaviour of the continuous-time weakly self-avoiding walk with contact self-attraction on $\\mathbb{Z}^4$, for sufficiently small attraction. We prove that the susceptibility and correlation length of order $p$ (for any $p>0$) have logarithmic corrections to mean field scaling, and that the critical two-point function is asymptotic to a multiple of $|x|^{-2}$. This shows that small contact self-attraction results in the same critical behaviour as no contact self-attraction; a collapse transition is predicted for larger self-attraction. The proof uses a supersymmetric representation of the two-point function, and is based on a rigorous renormalisation group method that has been used to prove the same results for the weakly self-avoiding walk, without self-attraction.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-10-26T23:30:51.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82B28",
      "82B27",
      "60K35"
    ],
    "keywords": [
      "four-dimensional weakly self-avoiding walk",
      "small contact self-attraction results",
      "critical behaviour",
      "rigorous renormalisation group method",
      "mean field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}