{
  "id": "2403.03040",
  "version": "v1",
  "published": "2024-03-05T15:20:50.000Z",
  "updated": "2024-03-05T15:20:50.000Z",
  "title": "The height gap of planar Brownian motion is $\\frac{5}π$",
  "authors": [
    "Antoine Jego",
    "Titus Lupu",
    "Wei Qian"
  ],
  "comment": "23 pages, 5 figures",
  "categories": [
    "math.PR",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We show that the occupation measure of planar Brownian motion exhibits a constant height gap of $5/\\pi$ across its outer boundary. This property bears similarities with the celebrated results of Schramm--Sheffield [18] and Miller--Sheffield [12] concerning the height gap of the Gaussian free field across SLE$_4$/CLE$_4$ curves. Heuristically, our result can also be thought of as the $\\theta \\to 0^+$ limit of the height gap property of a field built out of a Brownian loop soup with subcritical intensity $\\theta>0$, proved in our recent paper [3]. To obtain the explicit value of the height gap, we rely on the computation by Garban and Trujillo Ferreras [1] of the expected area of the domain delimited by the outer boundary of a Brownian bridge.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-03-05T15:20:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60J65",
      "60J67",
      "60J55"
    ],
    "keywords": [
      "planar brownian motion",
      "outer boundary",
      "constant height gap",
      "property bears similarities",
      "brownian loop soup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 23,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}