{
  "abstract": "We study cover times on discrete tori and ranges on $\\mathbb{Z}^d$ for elephant random walks with memory parameter $p\\in[0,1)$. In dimension one there is a phase transition at $p=3/4$. Away from criticality we determine the exact first-order asymptotics of the mean cover time, while at $p=3/4$ the mean is of order $L^2/\\sqrt{\\log L}$, a factor $\\sqrt{\\log L}$ larger than the natural fluctuation scale $L^2/\\log L$. We identify the limiting distribution under each of the three natural fluctuation normalizations. In dimensions $d\\geq2$, for every fixed $p<1$, the cover time has the same order as for simple random walk, in expectation and with high probability. The leading constants also agree when $p<(2d+1)/(4d)$, and at equality when $d\\geq3$. For the range on $\\mathbb{Z}^d$, we prove $L^1$ convergence to the simple random walk asymptotics when $d=2$ and $p<5/8$, and for every $p<1$ when $d\\geq3$. Finally, we give a cover-time upper bound for generalized step-reinforced random walks on finite groups in terms of a conditional $L^\\infty$ mixing profile.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.17264",
  "author_total": 1,
  "authors": [
    {
      "name": "Shuo Qin"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/aaj631eqjzg7f.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.17264",
  "fulltext_url": null,
  "id": "aaj631eqjzg7f",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.17264",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7213427080",
  "pdf_url": "https://arxiv.org/pdf/2609.17264",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/aaj631eqjzg7f.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:21:45Z",
      "native_id": "2609.17264",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.17264"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Cover times and ranges of elephant random walks",
  "title_text": "Cover times and ranges of elephant random walks",
  "type": "preprint",
  "updated_at": "2026-10-30T12:21:45Z",
  "url": "https://research.draconitic.com/paper/aaj631eqjzg7f",
  "venue": "arXiv",
  "versions": []
}