{
  "abstract": "In this paper, we study once-reinforced random walk on $\\mathbb{Z}$, and prove sharp Chung-type and Hartman-Wintner laws of the iterated logarithm for the range. Let $R_n$ denote the range at time $n$ and let $c>0$ be the reinforcement parameter. Then, almost surely, \\begin{equation*} \\liminf_{n\\to\\infty}\\frac{R_n}{\\sqrt{n/\\log\\log n}}=\\fracπ{\\sqrt{2}}, \\qquad \\limsup_{n\\to\\infty}\\frac{cR_n}{\\sqrt{2n\\log\\log n}}=1. \\end{equation*} The proofs are based on the independent increment decomposition of the inverse range and sharp generating function estimates.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.36790",
  "author_total": 3,
  "authors": [
    {
      "name": "Ze-Chun Hu"
    },
    {
      "name": "Renming Song"
    },
    {
      "name": "Li Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/1gap9a996276x.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.36790",
  "fulltext_url": null,
  "id": "1gap9a996276x",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.36790",
  "language": null,
  "licence": "CC-BY-4.0",
  "licence_label": "CC BY 4.0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/licenses/by/4.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7224095691",
  "pdf_url": "https://arxiv.org/pdf/2610.36790",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-27",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/1gap9a996276x.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:35:53Z",
      "native_id": "2610.36790",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.36790"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Laws of the iterated logarithm for the range of once-reinforced random walk on $\\mathbb{Z}$",
  "title_text": "Laws of the iterated logarithm for the range of once-reinforced random walk on ℤ",
  "type": "preprint",
  "updated_at": "2026-10-30T12:35:53Z",
  "url": "https://research.draconitic.com/paper/1gap9a996276x",
  "venue": "arXiv",
  "versions": []
}