{
  "id": "2207.09305",
  "version": "v1",
  "published": "2022-07-19T14:42:02.000Z",
  "updated": "2022-07-19T14:42:02.000Z",
  "title": "The wired minimal spanning forest on the Poisson-weighted infinite tree",
  "authors": [
    "Asaf Nachmias",
    "Pengfei Tang"
  ],
  "comment": "35 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study the spectral and diffusive properties of the wired minimal spanning forest (WMSF) on the Poisson-weighted infinite tree (PWIT). Let $M$ be the tree containing the root in the WMSF on the PWIT and $(Y_n)_{n\\geq0}$ be a simple random walk on $M$ starting from the root. We show that almost surely $M$ has $\\mathbb{P}[Y_{2n}=Y_0]=n^{-3/4+o(1)}$ and $\\mathrm{dist}(Y_0,Y_n)=n^{1/4+o(1)}$ with high probability. That is, the spectral dimension of $M$ is $\\frac{3}{2}$ and its typical displacement exponent is $\\frac{1}{4}$, almost surely. These confirm Addario-Berry's predictions in arXiv:1301.1667.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-19T14:42:02.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "wired minimal spanning forest",
      "poisson-weighted infinite tree",
      "simple random walk",
      "confirm addario-berrys predictions",
      "high probability"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 35,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}