{
  "id": "2111.00718",
  "version": "v2",
  "published": "2021-11-01T06:14:37.000Z",
  "updated": "2022-04-07T08:20:36.000Z",
  "title": "Spectral dimension of simple random walk on a long-range percolation cluster",
  "authors": [
    "Van Hao Can",
    "David A. Croydon",
    "Takashi Kumagai"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "Consider the long-range percolation model on the integer lattice $\\mathbb{Z}^d$ in which all nearest-neighbour edges are present and otherwise $x$ and $y$ are connected with probability $q_{x,y}:=1-\\exp(-|x-y|^{-s})$, independently of the state of other edges. Throughout the regime where the model yields a locally-finite graph, (i.e.\\ for $s>d$,) we determine the spectral dimension of the associated simple random walk, apart from at the exceptional value $d=1$, $s=2$, where the spectral dimension is discontinuous. Towards this end, we present various on-diagonal heat kernel bounds, a number of which are new. In particular, the lower bounds are derived through the application of a general technique that utilises the translation invariance of the model. We highlight that, applying this general technique, we are able to partially extend our main result beyond the nearest-neighbour setting, and establish lower heat kernel bounds over the range of parameters $s\\in (d,2d)$. We further note that our approach is applicable to short-range models as well.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-04-07T08:20:36.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K37",
      "35K05",
      "60J15",
      "60J35",
      "60J74",
      "82B43"
    ],
    "keywords": [
      "simple random walk",
      "spectral dimension",
      "long-range percolation cluster",
      "general technique",
      "on-diagonal heat kernel bounds"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}