{
  "abstract": "We study first-passage percolation on the $\\ell$-spread-out one-dimensional cycle of size $n$, where vertices are connected if their graph distance is at most $\\ell$. We assign i.i.d.~non-negative random weights from a Weibull distribution $ω_e \\sim \\mathrm{Exp}(1)^{1/θ}$ to the edges for $θ>0$ fixed. This paper investigates the transition in the asymptotic behavior of the passage time $T_n$ between two typical vertices and the hop-count of the optimal path as the connectivity parameter $\\ell$ diverges with $n$. We identify two fundamentally distinct geometric regimes. In the mesoscopic regime ($1 \\ll \\ell \\ll n$), the optimal path locally mimics a spatial branching random walk but remains globally constrained to a one-dimensional geometry. We establish a law of large numbers characterized by the front speed of a Crump--Mode--Jagers branching random walk, prove a central limit theorem with Gaussian fluctuations when $\\ell\\ll n^{1/4}$, and show that the expected hop-count grows proportionally with the spatial distance. In the macroscopic regime ($\\ell \\approx λn$ for $λ\\in (0,1/2)$), the graph becomes a highly connected mean-field network. We prove that the passage time collapses to a $\\log n$ scale with constant order non-Gaussian fluctuations, explicitly determining the extreme-value limit driven by the collision of two independent non-spatial CMJ processes. We establish a law of large numbers for the hop-count. Finally, we rigorously trace the transition in the order of the mean of $T_n$ between these two regimes, demonstrating an order transition for the passage time across the critical connectivity threshold $\\ell \\asymp n/\\log n$. Our results provide a comprehensive deterministic-range interpolation from spatial Gaussian fluctuations to mean-field extreme-value fluctuations.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.00110",
  "author_total": 2,
  "authors": [
    {
      "name": "Partha S. Dey"
    },
    {
      "name": "Daecheol Kim"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/c6c0sx53kvfq9.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.00110",
  "fulltext_url": null,
  "id": "c6c0sx53kvfq9",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.00110",
  "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": "W7206150601",
  "pdf_url": "https://arxiv.org/pdf/2609.00110",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/c6c0sx53kvfq9.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:04Z",
      "native_id": "2609.00110",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.00110"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    }
  ],
  "title": "Phase Transition and Fluctuation Results for First-Passage Percolation on Spread-Out Cycle Graphs",
  "title_text": "Phase Transition and Fluctuation Results for First-Passage Percolation on Spread-Out Cycle Graphs",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:04Z",
  "url": "https://research.draconitic.com/paper/c6c0sx53kvfq9",
  "venue": "arXiv",
  "versions": []
}