{
  "id": "2209.07283",
  "version": "v1",
  "published": "2022-08-23T12:01:42.000Z",
  "updated": "2022-08-23T12:01:42.000Z",
  "title": "On an extreme value law for the unipotent flow on $\\mathrm{SL}_2(\\mathbb{R})/\\mathrm{SL}_2(\\mathbb{Z})$",
  "authors": [
    "Maxim Kirsebom",
    "Keivan Mallahi-Karai"
  ],
  "comment": "13 pages, 5 figures, comments welcome!",
  "categories": [
    "math.DS",
    "math.NT",
    "math.PR"
  ],
  "abstract": "We study an extreme value distribution for the unipotent flow on the modular surface $\\mathrm{SL}_2(\\mathbb{R})/\\mathrm{SL}_2(\\mathbb{Z})$. Using tools from homogenous dynamics and geometry of numbers we prove the existence of a continuous distribution function $F(r)$ for the normalized deepest cusp excursions of the unipotent flow. We find closed analytic formulas for $F(r)$ for $r \\in [-\\frac{1}{2} \\log 2, \\infty)$, and establish asymptotic behavior of $F(r)$ as $r \\to -\\infty$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-08-23T12:01:42.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A17",
      "60G70",
      "11H06"
    ],
    "keywords": [
      "extreme value law",
      "unipotent flow",
      "normalized deepest cusp excursions",
      "extreme value distribution",
      "modular surface"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}