{
  "id": "2205.11468",
  "version": "v1",
  "published": "2022-05-23T17:11:41.000Z",
  "updated": "2022-05-23T17:11:41.000Z",
  "title": "Multiple points on the boundaries of Brownian loop-soup clusters",
  "authors": [
    "Yifan Gao",
    "Xinyi Li",
    "Wei Qian"
  ],
  "comment": "55 pages, 12 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "For a Brownian loop soup with intensity $c\\in(0,1]$ in the unit disk, we show that the almost-sure Hausdorff dimensions of simple and double points on the boundaries of its clusters are respectively equal to $2-\\xi_c(2)$ and $2-\\xi_c(4)$, where $\\xi_c(k)$ is the generalized disconnection exponent computed in arxiv:1901.05436. We further show that almost surely such points are dense on every portion of boundary of every cluster, when they exist. As an intermediate result, we establish a separation lemma for Brownian loop soups, which is a powerful tool for obtaining sharp estimates on non-intersection and non-disconnection probabilities in the setting of loop soups. In particular, it allows us to define a family of generalized intersection exponents $\\xi_c(k, \\lambda)$, and show that $\\xi_c(k)$ is the limit as $\\lambda\\searrow 0$ of $\\xi_c(k, \\lambda)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-05-23T17:11:41.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "brownian loop-soup clusters",
      "multiple points",
      "brownian loop soup",
      "almost-sure hausdorff dimensions",
      "intermediate result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 55,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}