Intermittent Cauchy walks enable optimal 3D search across target shapes and sizes
This work provides a rigorous foundation for the Lévy flight foraging hypothesis in 3D, revealing shape vulnerabilities driven by dimensionality, which is relevant for biological and engineered systems.
The paper tackles the problem of how target shape affects detectability in 3D random search, proving that Cauchy walks (Lévy exponent μ=2) achieve near-optimal detection across various target sizes and shapes, with expected detection time scaling as n/Δ for convex targets.
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 $μ\to 2$, 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.