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arXiv · 2606.04837

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 $M\leq N$, 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.

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Fabio Deelan Cunden, Noemi Cuppone, Giovanni Gramegna, Pierpaolo Vivo. 2026-06-03. Maximal Minimal Spacing for Random Points. https://doi.org/10.1007/s10955-026-03680-5

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