2026/03/11 by Matteo Stromieri, Emanuele Natale, Amos Korman · 1 voice
Computer Science · Mathematics · #cs.DS #math.PR
paper · pdf · doi:10.48550/arxiv.2603.10655
Target shape, not just size, plays a pivotal role in determining detectability during random search. We analyze intermittent Lévy walks in three dimensions, and mathematically prove that the widely observed Cauchy strategy (Lévy exponent μ= 2) uniquely achieves scale-invariant, near-optimal detection across a broad spectrum of target sizes and shapes. In a domain of volume n with boundary conditions, expected detection time for a convex target of surface area Δ optimally scales as n/Δ. Conversely, Lévy strategies with μ< 2 are slow at detecting targets with large surface area-to-volume ratios, while those with μ> 2 excel at finding large elongated shapes but degrade as targets become wider. Our results further indicate a continuous geometric transition: volume dictates detection near μ= 1, ceding dominance to surface area as μ→ 2, after which surface area and elongation couple to govern detection. Ultimately, 3D search introduces a pronounced sensitivity to target shape that is absent in lower dimensions. Our work provides a rigorous foundation for the Lévy flight foraging hypothesis in 3D by establishing the scale-invariant optimality of the Cauchy walk. Furthermore, our results reveal dimensionality-driven shape vulnerabilities and offer testable predictions for biological and engineered systems.