{
  "abstract": "We study the mixing time of a simple random walk on the small-world network defined by adding edges to $\\mathbb{Z}_n^d$ as follows: for each pair $\\{x,y\\}$ we add an edge with probability $Z_n/\\|x-y\\|^{d}$ with $Z_n$ chosen so that the average number of added edges to every vertex is $1$. When $d\\geq 3$, we show that with high probability the mixing time is of order $\\log n$ and that the random walk does not exhibit cutoff.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.12056",
  "author_total": 5,
  "authors": [
    {
      "name": "Zsuzsanna Baran"
    },
    {
      "name": "Jonathan Hermon"
    },
    {
      "name": "Anđela Šarković"
    },
    {
      "name": "Allan Sly"
    },
    {
      "name": "Perla Sousi"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/7az8wskrry9et.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.12056",
  "fulltext_url": null,
  "id": "7az8wskrry9et",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.12056",
  "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": "W7212527116",
  "pdf_url": "https://arxiv.org/pdf/2609.12056",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-15",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/7az8wskrry9et.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:29:54Z",
      "native_id": "2609.12056",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.12056"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Random walk on the small-world network model in 3 or more dimensions",
  "title_text": "Random walk on the small-world network model in 3 or more dimensions",
  "type": "preprint",
  "updated_at": "2026-10-30T12:29:54Z",
  "url": "https://research.draconitic.com/paper/7az8wskrry9et",
  "venue": "arXiv",
  "versions": []
}