{
  "abstract": "From $N+1$ random points on a line we wish to select $M+1$ points so as to maximize the minimal spacing between them. We consider an initial configuration with independent and identically distributed spacings. Equivalently, the points are arrival times of a generic renewal process. For general spacing distributions, and for all $M\\leq N$, we derive exact distributional identities for the maximal minimal spacing and obtain its asymptotic behavior. The problem admits a reformulation in terms of a threshold-resetting random walk. The walk advances by successive random increments and is reset to the origin upon exceeding a fixed threshold. The probability that the optimal spacing exceeds a given value coincides with the probability that the walk completes at least $M$ reset cycles within $N$ steps. This yields an exact representation in terms of first-passage functionals of the walk. The same mapping suggests a numerical scheme for the max-min spacing problem in the regime of large $N$ and $M$, whose accuracy is tested against the exact results obtained here.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": "Journal reference: Journal of Statistical Physics 193, 118 (2026)",
  "arxiv": "2606.04837",
  "author_total": 4,
  "authors": [
    {
      "name": "Fabio Deelan Cunden"
    },
    {
      "name": "Noemi Cuppone"
    },
    {
      "name": "Giovanni Gramegna"
    },
    {
      "name": "Pierpaolo Vivo"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/8a1t7ztx8ryfb.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2606.04837",
  "fulltext_url": null,
  "id": "8a1t7ztx8ryfb",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2606.04837",
  "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"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2606.04837",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-03",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/8a1t7ztx8ryfb.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:31Z",
      "native_id": "2606.04837",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2606.04837"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Maximal Minimal Spacing for Random Points",
  "title_text": "Maximal Minimal Spacing for Random Points",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:36Z",
  "url": "https://research.draconitic.com/paper/8a1t7ztx8ryfb",
  "venue": "arXiv",
  "versions": [
    {
      "doi": "10.1007/s10955-026-03680-5",
      "pid": null,
      "relation": "published-as",
      "title": null,
      "venue": null
    }
  ]
}