{
  "id": "2506.21516",
  "version": "v1",
  "published": "2025-06-26T17:39:18.000Z",
  "updated": "2025-06-26T17:39:18.000Z",
  "title": "On the visibility window for Brownian interlacements, Poisson cylinders and Boolean models",
  "authors": [
    "Yingxin Mu",
    "Artem Sapozhnikov"
  ],
  "comment": "11 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "We study visibility inside the vacant set of three models in $\\mathbb R^d$ with slow decay of spatial correlations: Brownian interlacements, Poisson cylinders and Poisson-Boolean models. Let $Q_x$ be the radius of the largest ball centered at $x$ every point of which is visible from $0$ through the vacant set of one of these models. We prove that conditioned on $x$ being visible from $0$, $Q_x/\\delta_{\\|x\\|}$ converges weakly, as $x\\to\\infty$, to the exponential distribution with an explicit intensity, which depends on the parameters of the respective model. The scaling function $\\delta_r$ is the visibility window introduced in arXiv:2304.10298, a length scale of correlations in the visible set at distance $r$ from $0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-06-26T17:39:18.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian interlacements",
      "visibility window",
      "poisson cylinders",
      "vacant set",
      "study visibility inside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}