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PreprintCC BY 4.0arXiv

Maximal Minimal Spacing for Random Points

  • Fabio Deelan Cunden
  • Noemi Cuppone
  • Giovanni Gramegna
  • Pierpaolo Vivo
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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≤ 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.

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Cite

BibTeXRIS
BibTeX
@misc{cunden2026maximal,
  title = {{Maximal Minimal Spacing for Random Points}},
  author = {Fabio Deelan Cunden and Noemi Cuppone and Giovanni Gramegna and Pierpaolo Vivo},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2606.04837},
  eprint = {2606.04837},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2606.04837},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Maximal Minimal Spacing for Random Points
AU  - Fabio Deelan Cunden
AU  - Noemi Cuppone
AU  - Giovanni Gramegna
AU  - Pierpaolo Vivo
PY  - 2026
DA  - 2026/09/03
PB  - arXiv
DO  - 10.48550/arxiv.2606.04837
UR  - https://arxiv.org/abs/2606.04837
AB  - 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≤ 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.
ER  -