{
  "id": "2502.07110",
  "version": "v1",
  "published": "2025-02-10T23:06:41.000Z",
  "updated": "2025-02-10T23:06:41.000Z",
  "title": "Limit distributions for cycles of random parking functions",
  "authors": [
    "J. E. Paguyo",
    "Mei Yin"
  ],
  "comment": "19 pages, 2 figures, comments welcome!",
  "categories": [
    "math.PR",
    "math.CO"
  ],
  "abstract": "We study the asymptotic behavior of cycles of uniformly random parking functions. Our results are multifold: we obtain an explicit formula for the number of parking functions with a prescribed number of cyclic points and show that the scaled number of cyclic points of a random parking function is asymptotically Rayleigh distributed; we establish the classical trio of limit theorems (law of large numbers, central limit theorem, large deviation principle) for the number of cycles in a random parking function; we also compute the asymptotic mean of the length of the $r$th longest cycle in a random parking function for all valid $r$. A variety of tools from probability theory and combinatorics are used in our investigation. Corresponding results for the class of prime parking functions are obtained.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2025-02-10T23:06:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "60C05",
      "60F05"
    ],
    "keywords": [
      "limit distributions",
      "cyclic points",
      "large deviation principle",
      "th longest cycle",
      "central limit theorem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}