{
  "id": "2006.06532",
  "version": "v1",
  "published": "2020-06-11T15:43:33.000Z",
  "updated": "2020-06-11T15:43:33.000Z",
  "title": "Kotani's Theorem and the Lace Expansion",
  "authors": [
    "Gordon Slade"
  ],
  "comment": "12 pages",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In 1991, Shinichi Kotani proved a theorem giving a sufficient condition to conclude that a function $f(x)$ on ${\\mathbb Z}^d$ decays like $|x|^{-(d-2)}$ for large $x$, assuming that its Fourier transform $\\hat f(k)$ is such that $|k|^{2}\\hat f(k)$ is well behaved for $k$ near zero. We prove an extension of Kotani's Theorem and combine it with the lace expansion to give a simple proof that the critical two-point function for weakly self-avoiding walk has decay $|x|^{-(d-2)}$ in dimensions $d>4$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-06-11T15:43:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "42B05",
      "82B41",
      "82B27",
      "60K35"
    ],
    "keywords": [
      "kotanis theorem",
      "lace expansion",
      "shinichi kotani",
      "fourier transform",
      "sufficient condition"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 12,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}