{
  "abstract": "For vertex-transitive Ramanujan graphs with logarithmic girth, a simple random walk bridge of length of order $\\log N$, where $N$ is the size of the graph, has a maximum distance that changes from order $\\sqrt{\\log N}$ to order $\\log N$ in a bounded critical window. We prove this by separating bridges whose lifts to the regular tree close from those whose lifts do not. A uniform two-term return estimate determines the probabilities of these two cases and the real-valued critical center. In the same $O(1)$ window, the normalized range has a two-point limiting law whose mixture weights vary nontrivially across the window.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.35213",
  "author_total": 1,
  "authors": [
    {
      "name": "Itai Benjamini"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/2j0arwds7dkkn.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.35213",
  "fulltext_url": null,
  "id": "2j0arwds7dkkn",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.35213",
  "language": null,
  "licence": "CC0-1.0",
  "licence_label": "CC0",
  "licence_source": "article",
  "licence_url": "http://creativecommons.org/publicdomain/zero/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.CO"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "2610|Stochastic processes and statistical mechanics"
    ]
  ],
  "openalex": "W7214752091",
  "pdf_url": "https://arxiv.org/pdf/2609.35213",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/2j0arwds7dkkn.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:25:17Z",
      "native_id": "2609.35213",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.35213"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.CO",
      "subfield": "combinatorics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Very sharp distance and range transitions for random walk bridges on Ramanujan graphs",
  "title_text": "Very sharp distance and range transitions for random walk bridges on Ramanujan graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:25:17Z",
  "url": "https://research.draconitic.com/paper/2j0arwds7dkkn",
  "venue": "arXiv",
  "versions": []
}