{
  "abstract": "Rare events and large-deviation theory have attracted considerable interest in the study of random walks. Here, we investigate the rare events of the positional distribution of continuous-time random walks under ordinary and equilibrium initial conditions using the saddle point method. We show that the far-tail behavior is sensitive to the expansion of the waiting-time distribution near its minimum value. For a general power-series expansion near the minimum waiting time, the positional distribution exhibits an approximately exponential decay in its far tails. In contrast, when the waiting-time distribution has a cutoff at its minimum value, the positional distribution exhibits an unexpected Gaussian decay, in contrast to the far-tail behavior of the renewal-count distribution. We further find that the most likely number of renewals required to reach a given position, as well as the associated rate functions, is insensitive to the initial conditions, whereas the far tails of the positional distribution retain a dependence on the initial conditions. Our results provide new insights into the rare-event statistics and large deviations of random walks.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38492",
  "author_total": 5,
  "authors": [
    {
      "name": "Chen Zhou"
    },
    {
      "name": "YuDa Cheng"
    },
    {
      "name": "Yangji Li"
    },
    {
      "name": "Jing Xu"
    },
    {
      "name": "Wanli Wang"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/by8fh4q0tc9xa.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38492",
  "fulltext_url": null,
  "id": "by8fh4q0tc9xa",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38492",
  "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",
      "cond-mat.stat-mech"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38492",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/by8fh4q0tc9xa.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.38492",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38492"
    }
  ],
  "subjects": [
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Far Tails of the Positional Distribution in random walk: A Saddle-Point Approach",
  "title_text": "Far Tails of the Positional Distribution in random walk: A Saddle-Point Approach",
  "type": "preprint",
  "updated_at": "2026-11-01T12:20:44Z",
  "url": "https://research.draconitic.com/paper/by8fh4q0tc9xa",
  "venue": "arXiv",
  "versions": []
}