{
  "abstract": "We determine the limiting distribution of the cover time of simple random walk on the two-dimensional discrete torus. For the continuous-time walk with total jump rate one on $(\\mathbb Z/N\\mathbb Z)^2$, let $T_N$ denote its cover time. We prove that $\\frac{T_N}{(2/π)N^2\\log N}-2\\log N+\\log\\log N \\Longrightarrow G+\\log(κZ)$, where $G$ is a standard Gumbel random variable, $Z$ is the total mass of the critical Gaussian multiplicative chaos associated with the zero-average Gaussian free field on the unit torus, $G$ and $Z$ are independent, and $κ>0$ is deterministic. This answers the limit-law question suggested by Aldous and recorded by Dembo, Peres, Rosen and Zeitouni. The proof identifies the random fluctuations in the number of small, well-separated unvisited components at a deterministic time before coverage, and then estimates the time needed to visit the remaining components.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.01852",
  "author_total": 1,
  "authors": [
    {
      "name": "Yechi Zhou"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/84v4sx7d4q4f7.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.01852",
  "fulltext_url": null,
  "id": "84v4sx7d4q4f7",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.01852",
  "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"
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  ],
  "openalex": "W7216021107",
  "pdf_url": "https://arxiv.org/pdf/2610.01852",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/84v4sx7d4q4f7.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:26:07Z",
      "native_id": "2610.01852",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.01852"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "A limit law for the cover time of the two-dimensional discrete torus",
  "title_text": "A limit law for the cover time of the two-dimensional discrete torus",
  "type": "preprint",
  "updated_at": "2026-10-30T12:26:07Z",
  "url": "https://research.draconitic.com/paper/84v4sx7d4q4f7",
  "venue": "arXiv",
  "versions": []
}