{
  "id": "2107.14347",
  "version": "v1",
  "published": "2021-07-29T22:01:20.000Z",
  "updated": "2021-07-29T22:01:20.000Z",
  "title": "Subcritical Connectivity and Some Exact Tail Exponents in High Dimensional Percolation",
  "authors": [
    "Shirshendu Chatterjee",
    "Jack Hanson",
    "Philippe Sosoe"
  ],
  "comment": "63 pages, 6 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "In high dimensional percolation at parameter $p < p_c$, the one-arm probability $\\pi_p(n)$ is known to decay exponentially on scale $(p_c - p)^{-1/2}$. We show the same statement for the ratio $\\pi_p(n) / \\pi_{p_c}(n)$, establishing a form of a hypothesis of scaling theory. As part of our study, we provide sharp estimates (with matching upper and lower bounds) for several quantities of interest at the critical probability $p_c$. These include the tail behavior of volumes of, and chemical distances within, spanning clusters, along with the scaling of the two-point function at \"mesoscopic distance\" from the boundary of half-spaces. As a corollary, we obtain the tightness of the number of spanning clusters of a diameter $n$ box on scale $n^{d-6}$; this result complements a lower bound of Aizenman.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-07-29T22:01:20.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60K35",
      "82B43",
      "82B27"
    ],
    "keywords": [
      "high dimensional percolation",
      "exact tail exponents",
      "subcritical connectivity",
      "lower bound",
      "spanning clusters"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 63,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}