{
  "abstract": "We study four paths that join the starting site to the final position of a simple random walk of \\(N=L^2\\) steps on the infinite cubic lattice. The paths are selected on the walk trace---the graph of visited sites and traversed bonds---by different rules: the undirected shortest path \\(S_1\\), the directed shortest path \\(S_2\\), the time-respecting shortest path \\(S_3\\), and the depth \\(D_e\\) of the final site in a depth-first-search tree. Monte Carlo simulations for \\(L\\) up to 32768 give mean lengths scaling as \\(L^{d_O}\\), with \\(d_{S_1}=1.48005(9)\\), \\(d_{S_2}=1.49868(5)\\), \\(d_{S_3}=1.51205(5)\\), and \\(d_{D_e}=1.7099(7)\\), so that \\(d_{S_1}<d_{S_2}<d_{S_3}<d_{\\rm LE}<d_{D_e}\\), where $d_{\\rm LE}=1.62400(5)$ is the known loop-erased-random-walk exponent. Imposing traversal directions increases the shortest-path exponent, and imposing time order increases it further; these constraints therefore change the scaling exponent, not only the amplitude. For every path, the standard deviation scales with the same exponent as the mean, so it is scale-invariant, unlike the visited-site mass; the rescaled length distributions collapse onto a finite-width scaling function. No analytical theory currently predicts these exponents; our results provide precise numerical benchmarks for future studies of path geometry on random-walk traces.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.39612",
  "author_total": 3,
  "authors": [
    {
      "name": "Qiyuan Shi"
    },
    {
      "name": "Xinyi Li"
    },
    {
      "name": "Youjin Deng"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/eqv71erf8kphr.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.39612",
  "fulltext_url": null,
  "id": "eqv71erf8kphr",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.39612",
  "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",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.39612",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-29",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/eqv71erf8kphr.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.39612",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.39612"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Fractal dimensions of paths on three-dimensional random-walk traces",
  "title_text": "Fractal dimensions of paths on three-dimensional random-walk traces",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:50Z",
  "url": "https://research.draconitic.com/paper/eqv71erf8kphr",
  "venue": "arXiv",
  "versions": []
}