Maximal Minimal Spacing for Random Points

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. The problem is equivalent to optimally grouping consecutive gaps into $M$ blocks and maximizing the smallest block sum. For general gap distributions, and for all $M\leq N$, we derive exact distributional identities for the optimal 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.