J Stat Phys. 2026;193(9):118. doi: 10.1007/s10955-026-03680-5. Epub 2026 Aug 27.
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.
PMID:42666375 | PMC:PMC13522019 | DOI:10.1007/s10955-026-03680-5
Nevin Manimala