{
  "abstract": "We study first-passage percolation on the $\\ell$-spread-out line graph, where each vertex $i\\in\\{0,1,\\dots,n\\}$ is connected to all others at distance at most $\\ell$. Here, we focus on the microscopic regime, with $\\ell$ fixed as $n\\to\\infty$. Independent nonnegative weights are assigned to these edges. We obtain a law of large numbers and precise fluctuation results for the passage time $T_n$ from $0$ to $n$. If the weight distribution has finite variance or a heavy tail with exponent above $2/\\ell_c$ where $\\ell_c=\\ell(\\ell+1)/2$, then $T_n$ satisfies a Gaussian CLT with $\\sqrt{n}$ scaling. In contrast, for heavier-tailed distributions, with index below the threshold, we show that $T_n$, appropriately centered and scaled, converges to a non-Gaussian stable law. We also prove an LLN and CLT for the number of edges in the minimizing path. The key tool is a pivot-node decomposition; the geodesic can be segmented into i.i.d. blocks, leading to a renewal structure. Our results extend the classical one-dimensional CLT to include finite-range connectivity and heavy tails, revealing a new distributional phase transition in the fluctuations of $T_n$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2607.28865",
  "author_total": 2,
  "authors": [
    {
      "name": "Partha S. Dey"
    },
    {
      "name": "Daecheol Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/0sxp2vyqyxw8m.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2607.28865",
  "fulltext_url": null,
  "id": "0sxp2vyqyxw8m",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2607.28865",
  "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": "W7172272987",
  "pdf_url": "https://arxiv.org/pdf/2607.28865",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-02",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/0sxp2vyqyxw8m.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:20Z",
      "native_id": "2607.28865",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2607.28865"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "First-Passage Percolation on Spread-out line graphs: Microscopic Regime",
  "title_text": "First-Passage Percolation on Spread-out line graphs: Microscopic Regime",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/0sxp2vyqyxw8m",
  "venue": "arXiv",
  "versions": []
}