{
  "id": "2410.20776",
  "version": "v1",
  "published": "2024-10-28T06:37:59.000Z",
  "updated": "2024-10-28T06:37:59.000Z",
  "title": "Scaling limit for the cover time of the $λ$-biased random walk on a binary tree with $λ<1$",
  "authors": [
    "David A. Croydon"
  ],
  "comment": "21 pages",
  "categories": [
    "math.PR"
  ],
  "abstract": "The $\\lambda$-biased random walk on a binary tree of depth $n$ is the continuous-time Markov chain that has unit mean holding times and, when at a vertex other than the root or a leaf of the tree in question, has a probability of jumping to the parent vertex that is $\\lambda$ times the probability of jumping to a particular child. (From the root, it chooses one of the two children with equal probability.) For this process, when $\\lambda<1$, we derive an $n\\rightarrow \\infty$ scaling limit for the cover time, that is, the time taken to visit every vertex. The distributional limit is described in terms of a jump process on a Cantor set that can be seen as the asymptotic boundary of the tree. This conclusion complements previous results obtained when $\\lambda\\geq 1$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-10-28T06:37:59.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05C81",
      "60J27",
      "60J50",
      "60J75"
    ],
    "keywords": [
      "biased random walk",
      "cover time",
      "binary tree",
      "scaling limit",
      "continuous-time markov chain"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}